First, let's correct this...
The Taffy units fought bravely, until they had run out of serviceable aircraft
This is incorrect. Taffy did not run out of aircraft, Kurita turned away. After Kurita turned away, aircraft were still returning and air attack continued. Yamato and Chikuma were attacked by Avengers from Taffys 2 and 3.
Probably not effectively or timely.
tl;dr Halsey receives news of the danger at 0800. By 0930 Kurita is already withdrawing. At best a one-way strike could be organized to attack Kurita as he withdraws. Communications between 3rd and 7th Fleet are extremely poor, organizing a landing on Taffys carriers is almost impossible. Assuming they have the range, they could land on freshly captured Leyte airfields. However, these airfields are in very rough shape and under infantry attack; there will be no second strike.
Let's answer this in two parts.
My understanding is that Halsey's fleet was too far away to conduct a "round trip" attack on Kurita's fleet. My hypothesis was that Halsey would fly in enough planes on "one way" trips to make up for the Taffy losses, those planes would refuel on the Taffy carriers, and then make their strikes. Then is this logistically possible[?]
Second, what impact would such a strike have had on the battle?
Before we begin, some important context.
Communications between fleets was crap.
Communications between fleets (really between everyone) in WW2 were slow and complicated. Messages from Taffy had to be dictated, encrypted, transmitted to 7th Fleet, received, decrypted, considered by 7th Fleet staff, and then a message to 3rd Fleet considered, dictated, encrypted, transmitted, received, decrypted, considered by 3rd Fleet, and handed up the ranks.
The messages were short, used jargon, and nonsense to throw off code breakers making plenty of opportunity for misunderstandings. This all had to be done carefully and according to strict protocols to avoid miscommunications. We're proposing to do this by hand, in combat.
For example, the simple message asking where the fleet carriers were...
Where is, repeat, where is Task Force Thirty Four?
Was encoded like this...
TURKEY TROTS TO WATER GG FROM CINCPAC ACTION COM THIRD FLEET INFO COMINCH CTF SEVENTY-SEVEN X WHERE IS RPT WHERE IS TASK FORCE THIRTY FOUR RR THE WORLD WONDERS.
And was read by Halsey as...
Where is, repeat, where is Task Force Thirty Four? The world wonders.
Who took it as a personal insult.
Keep this in mind while we try to coordinate a one-way scratch carrier strike between two fleets hundreds of miles apart in the middle of the largest naval battle in history.
Could 3rd Fleet launch a one-way strike?
The idea is that 3rd Fleet is too far away to make a round trip strike, but maybe they could send a strike and that strike could then land on a Taffy carrier or airfield.
A lot of details have to be filled in before this can be answered.
Did 3rd Fleet aircraft have the range to strike Kurita and land safely?
Kurita was between 3rd Fleet and the Taffys and Leyte, so if they can reach the Taffys or Leyte they can reach Kurita. Did they have the range? Avenger strike aircraft had considerable range, but with how much ordinance?
My very rough estimate is about 800 km. This is well inside the range of an Avenger. I'll leave this for someone else to figure out the details because it's the least interesting part of the analysis. Either they do or they don't. I'll assume they do.
When would the strike arrive?
What is the earliest they could have learned there was a problem? When did 3rd Fleet know there was a crisis?
Halsey claims he did not receive word that 7th Fleet had a problem until at least 0800 over an hour after Kurita attacked, and possibly as late as 1000 after Kurita had already turned away. The confusion could be Halsey covering his ass as to why he ignored their cries for help.
Let's say Halsey receives the distress calls around 0800 and springs into action! Now he has to organize a strike. 3rd Fleet has been launching strikes at Ozawa since morning. Presumably he could order the next strike to instead launch against Kurita.
The strike would have be be refueled to have additional range. It's possible this would put them over their maximum takeoff weight in which case armaments would have to be removed or replaced. Deck handlers would have to scramble to shift plans around for the new plan.
Pilots would have to be briefed. What were the targets? Where are the targets? What is their course and speed? Where are they landing? If it's a carrier, what is its course and speed and will it be there when they land? How long does it take for 3rd Fleet to get this information from 7th Fleet?
Launching the strike and forming up takes considerable time. They would then have to fly to Kurita, attack, and land. With a cruising speed of about 350 km/h, it would take over 2 hours to cover the roughly 800km distance.
Even if the strike magically launched immediately upon receiving the distress call at 08:00, like a video game, it would not reach Taffy 3 until after 10:00. By this time the battle was over and Kurita was withdrawing. In reality it would have taken significantly longer.
Could they land on Taffy carriers?
This gets very complicated and risky.
Did 3rd Fleet even know the position and disposition of the Taffys? Probably not. Why would they? That's 7th Fleet's job. Could they have found out? Sure, but it would likely take hours as the request goes from 3rd Fleet to 7th Fleet to the Taffys who probably don't give it priority because they're all under attack or conducting intense air operations.
Let's say 3rd Fleet somehow gets accurate positions, courses, and speeds for the Taffys at the time of launching the strike. Will those still be accurate hours later when the strike arrives? What if they change course and speed? Taffy 1 has to evade air attack. Taffy 2 changes course to avoid Kurita. Taffy 3 is evading wildly. They're all conducting air operations which requires turning into the wind.
The Taffys are all undergoing their own air operations, strikes, combat air patrols, and recovery. How do they coordinate receiving the strike from 3rd Fleet? What happens if the strike shows up and they are unprepared to receive the strike? Maybe they're under attack, their decks are not clear, they're damaged... If so, the 3rd Fleet strike might run out of fuel and drop into the sea.
Probably don't try to land on the Taffy carriers. It's risky and complicated and requires a lot of coordination and the carriers might not be where they say they will be, or damaged, on fire, or sunk.
Land on an airfield
If you're throwing together a hasty one-way strike mission, it's a lot easier if your airfield isn't moving, on fire, or under water.
Dulag and Tacloban airfields had been taken on Leyte a few days earlier on 20 October, but had been under continuous Japanese air attack. These would be the logical and simple choice to receive and refuel the strike.
Some Taffy aircraft diverted to Dulag that morning. Upon landing they were handed carbines and told they were under attack by Japanese infantry. So while the strike could land, it is doubtful they would be refueled and rearmed in time to make a second go.
Could a hasty one-way 3rd Fleet strike have effected the battle?
No, there just wasn't enough time. By the time 3rd Fleet's strike was organized, launched, and arrived the battle was long over.
The US realized they were under attack at roughly 06:50. By 07:00 Yamato opens fire on the escort carriers at 31 km. Around 08:00 Halsey receives the distress calls. At 09:10 Kurita orders a regroup to the north and then a full withdrawal.
This leaves a little over an hour for 3rd Fleet to organize and launch a strike, and for it to fly to the battle. There simply isn't time.
3rd Fleet might have been able to attack Kurita as he withdrew, but this would not effect the outcome of the battle. It would be questionable to put pilots' lives on such a risky attack against a retreating enemy.
What if Kurita attacks?
Historically, upon learning the rest of Operation Shō-Gō 1 had failed Kurita withdrew. What if he went all-in?
By 09:20 Taffy 3 is combat ineffective. The heroic delaying action earlier in the battle was conducted by well-armed, high-speed fleet destroyers. These are now crippled and sinking. The three remaining destroyer escorts are designed for convoy duty; slow and poorly armed (two 5" guns and three torpedo tubes each), combined their firepower was barely equal to a single fleet destroyer.
In contrast the Japanese have four battleships, two serviceable heavy cruisers (Haguro and Tone), two light cruisers, and eleven destroyers. All expressly designed to sink ships. The Japanese destroyers alone could handle what remains of Taffy 3.
Japanese gunners have been firing at Taffy 3 and closing range for almost two hours. At the beginning of the battle the unarmored US ships were benefiting from the Japanese mistaken belief they were firing at larger and armored ships; their armor piercing shells were passing through without exploding. By this point in the battle, Japanese gunners were beginning to realize they were firing at unarmed targets and were switching to high-explosive to devastating effect. Without the heroic antics of the fleet destroyers to distract them, Japanese gunners make short work of the escort vessels.
Survivors are mopped up by the historical Japanese air attack at 10:50.
Kurita moves on to bombard the landings. The 3rd Fleet strike arrives either while is on his way to bombard the landings or on his way out. The hasty strike would struggle to sink even a single battleship; HMS Renown and Repulse took nearly 100 aircraft to sink them. Given the intensity of air attack Kurita has endured already, this one is unlikely to deter him.
Upon landing at the Leyte airfields, the pilots will find they are under Japanese infantry attack and under threat of bombardment by Kurita. A second strike is unlikely.