According to the 1940 census, there were 126,948 Japanese and Japanese Americans in the contiguous United States (so excluding Hawaii).
Geographically, the so-called evacuation orders only applied to California, Oregon, Washington and the southern portion of Arizona. The orders did not apply to Japanese in the other 44 states or other territories. However, most (112,354) Japanese in the contiguous states lived in California, Oregon and Washington.
So considering the base population affected to be 112,354, the main way people avoided being detained was to move eastward when (immediately after DeWitt's proclamation #1) people were free to do so.
As a specific example, I quote Japanese-American Junkoh Harui:
The government decreed that those who lived east of the Cascade
Range did not have to go to camp because it was a non-military vital area. Most of
the vital areas were all west of the Cascade Range. You know, the airports and the
manufacturing, shipbuilding, etc., were all on the West Coast, so for instance, those
people who lived in Spokane didn’t have to go to camp. So what happened was
somebody found out about that and said, let’s move to Moses Lake, so 3 families
from Bainbridge Island moved to Moses Lake prior to the date that was scheduled for
evacuation to camps. And we missed it by 2 days. We left 2 days before they did.
So he moved within Washington state, away from the coast, and thereby avoided having to go to a camp.
Nonetheless, in total 111,155 people were taken to relocation centers, which is almost the full 112,354. (Of these 29,516 were allowed to move out of the camps prior to December 1944 to work or attend college).
Main source of above is BACKGROUND TO JAPANESE AMERICAN RELOCATION on the Central Washington University server.