165-15s are probably the most common replacement for the original 5.90-15s. They are actually a little smaller in overall diameter than the originals, so a lot of folks go with 185-15s. I would regard this as the largest practical size.
I can't agree that 165s are hard to find. There seem to be several including Vredesein and Kumho. I have 185-15 Vreds on my Jag, and they are GREAT!
Wanting the extra diameter, but not the extra width, I have gone with 165-400 tires on Dayton 16 x 6 rims (VERY tight fit!). I doubt that 165-400s or 165-16s are even available these days, certainly not at a reasonable price.