I wonder if it is bacterial spot. The yellow ring around the
black spot is consistent with this description of bacterial spot. Reading the
link shows that there are some sprays you could try.

http://www.plantprobs.net/plant/tomato/bacterial-diseases/bacterial-spot.html

However, if you have the garden space I wonder if it would
be worth starting a new crop far away from this one. You don’t want this
problem moving between your potatoes and tomatoes again next season if it stays
in the soil. Crop rotation is going to be important.

Thanks for the reply, shift. I responded to your post but my reply never made it on to the site. :(

Unfortunately, I don't have much space to garden. I have three 4'x8' beds, so even crop rotation is a bit of a challenge -- I can't go long before I have to repeat a crop in a bed. After 15 years, I think I've probably got a lot of soil-borne diseases lurking. I tried solarizing the beds last summer, not sure it did much. I wonder if I should just start completely over from scratch and replace the soil in these growing beds? I wonder if that would help significantly?