He was quite a bit involved until 1943 in minute details. We can conclude this from all the orders that we know of, however sometimes circumstantial they might look.
For the later years when the camps reached their highest rates of killings, I don't know enough about the details he may have known or not. Perhaps he found a new thing to be per-occupied about, or considered it done?
Q Were there additional verbal or written orders from Hitler considering the details of the extermination campaign?
No. As there are no written orders from Hitler 'ordering the Holocaust' at all, the premise of 'additional' is a bit flawed.
There are some indirect and direct clues to be found in Goebbels diaries for example. But they remain vague and superficial. Or in orders to officers, mostly concerning surrounding details.
And considering the details, it might be worth reading the following with the question about details in mind:
I am now referring to the evacuation of the Jews, the extermination of the Jewish people. It's one of those things that is easily said: 'The Jewish people are being exterminated', says every party member, 'this is very obvious, it's in our program, elimination of the Jews, extermination, we're doing it, hah, a small matter.' […]
But none has observed it, endured it. Most of you here know what it means when 100 corpses lie next to each other, when there are 500 or when there are 1,000. To have endured this and at the same time to have remained a decent person — with exceptions due to human weaknesses — has made us tough, and is a glorious chapter that has not and will not be spoken of. Because we know how difficult it would be for us if we still had Jews as secret saboteurs, agitators and rabble-rousers in every city, what with the bombings, with the burden and with the hardships of the war. If the Jews were still part of the German nation, we would most likely arrive now at the state we were at in 1916 and 17 […]
–– Himmler, Posen speeches,
Indicating that even for many SS members the methods and amounts of killing was not always easy to stomach. Interest in overt details is perhaps overrated, especially if a plan is that gross and monstrous.
Q Did he receive reports on its progress?
We might asume that with some reason, see below, but we have again no direct written evidence for it. e know that Heydrich showed him at least the territorial aspects of where the camps were planned to be operating in 1941. But at that stage probably still envisioning to transfer all Jews into the to conquer Soviet Union territory.
For example Hans Frank pestered Hitler about his 'Jewish problem' in the Generalgouvernement in June 1941, to which Hitler replied he shouldn't worry, as they all about to be "removed" anyway. These exchanges between Hitler and underlings are quite numerous and well documented.
Since Himmler was always plotting for more power he made sure that what he did was approved by Hitler. Those killings done immediately behind the front by Wehrmacht, police units and SS, both of the later now under Himmler's control, a few satistics will have made it into Adolf's ears.
For the statistics of the Einsatzgruppen shootings it is documented how he radioed commands to them demanding to be kept informed on the 'progress'.
Q Did he visit any death camps?
This was not passed down to us.
Himmler visited. On 17 July with a detailed inspection of the killing process. On the next best social occasion of dinner he spoke in such a manner that those present concluded that now the Germans would be determined to kill all Jews. One of the attendees sent this word to Switzerland so that the world would know.
Q Did he mention the Final Solution in his later speeches?
Oh, yes he did.
According to Christian Gerlach: "Die Wannsee-Konferenz, das Schicksal der deutschen Juden und Hitlers politische Grundsatzentscheidung, alle Juden Europas zu ermorden", WerkstattGeschichte 18, Ergebnisse Verlag, Harnburg 1997, p 7-44 (PDF), it is now pretty sure that the order was given officially in a private surrounding on December 12, 1941, Wannseekonferenz was planned earlier as a meeting, but postponed to January, to allow for new Führer's wishes and orders to authorise it.
Then Hitler took the opportunity on January 30 to announce in the broadest possible daylight:
I do not want to miss pointing out what I pointed out on 3rd of September  in the German Reichstag, that if Jewry were to plunge the world into war, the role of Jewry would be finished in Europe. They may laugh about it today, as they laughed before about my prophecies. The coming months and years will prove that I prophesied rightly in this case too. (src)
Ten days after the Wannsee Conference, on 30 January, Hitler declared in his speech on the occasion of the anniversary of the "seizure of power" in the Sportpalast: "We are aware that the war can only end with either the extermination of the Aryan peoples or the disappearance of Judaism from Europe". In his statement on the occasion of the 22nd anniversary of the founding of the party on 24 February 1942, he stated that "my prophecy will be fulfilled that this war will not annihilate Aryan humanity, but will exterminate the Jew".
–– Peter Longerich: "Wannseekonferenz. Der Weg zur »Endlösung«", Pantheon: München, 2016.
That he knew quite a lot of details about the murders is evidenced by Aktion T4. Killings, also with gas already, ordered by Hitler to begin and later ordered by him to be stopped.
Hitler also interfered with orders between Göring and Speer, according to Goebbels, to a shift from the slave labour with Jews in Germany, which Göring favoured, to instructing Speer to deport them all.
In May 1942 Himmler met Hitler for a few days to discuss 'things' – we have no documents. One topic in these days seems obvious, as the extermnation policy radicalised again dramatically following that meeting.
On 11, 12, 14 of July Himmler repeatedly sought Hitler's authorisation to for more transport capacity into the extermination camps. He got it.
This is of course all complicated by intentionally vague and deceptive language in much of the surviving evidence. Many files were intentionally destroyed.
As a batch of documents shows, Hitler announced his decision in principle to murder all of Europe’s Jews on or around December 12, 1941. He gave a speech during a meeting of about 50 regional and sectional leaders of the Nazi Party on that day, about which Goebbels noted:
Regarding the Jewish question the Führer is determined to clear the table. He warned the Jews that if they were to cause another world war, it would lead to their own destruction. Those were not empty words. Now the world war has come. The destruction of the Jews must be its necessary consequence. We cannot be sentimental about it.
According to Goebbels, Hitler added – in reference to German military losses – that those responsible for the war “will have to pay for it with their own lives.”58 On December 16, Hans Frank, who had been present at Hitler’s speech, told his administration in Kraków that it was necessary to “liquidate” the Polish Jews in Poland since, according to his conversations with those in charge of the Reichskommissariate, it would not be possible to deport them eastward to Soviet territory. (In October, Frank had been told that such deportations might be possible later.) Frank said that instead – referring back to Hitler’s remarks four days earlier – an organization was to be established in the General Government in cooperation with authorities in the Reich which would, in ways yet to be determined, produce a “destructive result” (Vernichtungserfolg) against the Jews. The Jews had to disappear during the war because Nazi thinking demanded it and because they were “useless eaters.”59 On December 14, Rosenberg, the minister for the occupied Soviet territories, told Hitler that “now, after the decision” he did not want to talk publicly about the extermination of the Jews. According to Rosenberg, Hitler agreed with him, adding that the Jews had to bear the consequences of having brought destruction upon the German people.
Through the RSHA, Himmler af rmed in May 1943 that all Jews – including the frail, armament workers and members of the Reich Association of German Jews – should be deported to the ‘east’ or Theresienstadt by June 30, 1943.35 Hitler had energetically reemphasized in another meeting of Nazi Party leaders on May 7 that all of the Jews in Europe were to be exterminated, which triggered activities by others.
–– Christian Gerlach: "The Extermination of the European Jews", New Approaches to European History, Cambridge University Press: Cambridge, New York, 2016.
It is also quite revealing to compare a genuine written order, a Führererlass from 1944 with History of the Jews in Hungary – Germany invades Hungary
Q Was he discussing it in his infamous table talks?
We do not know whether he discussed this in his 'table talks'. All those books that are called "Table Talk" do not mention this directly.
There is a mention of a former and discarded plan of deporting Jews to the East on May 12, 1942 in which he ponders how useless it would be to resettle the Jews to arctic deserts, as they would survive this…
But the reliability of these edited books is severely limited anyway.