No doubt someone will correct me if I'm wrong, but I'm pretty sure neither tomatoes nor herbs will do their best in less than full sun. And by full sun, I mean at least 5 hours. So many herbs come from the mediterranean where sunshine is constant.

Is there any way you can put pots in sunnier spots in your yard, such as around the pool? Herbs do very well in pots and so do tomatoes. I grew both full size and cherry tomatoes in pots last summer and they did great.

I would like to second kcbarbara's suggestion. I have a sunny garden, but I also like to plant tomatoes and peppers in large pots on my patio. Just as soon as we can get rid of the SNOW in the Chicago area!!!, I will be planting radishes, lettuce, and spinach in pots that will later hold flowers. We really do love these early veggies that I can plant before the garden is tillable
Jean

Maybe asparagus? As you said, salad greens--lettuces, spinach, and chard would probably do okay. I'd try cooler-weather veggies, peas, cucumbers, and maybe broccoli and/or cauliflower--you might not get as much as you would in full sun, but if you plant them a little later in the season, maybe you can still get a good crop. Do you have anything you can use to reflect sunlight onto the garden (indirect light) for another hour or two? That might help with those plants that prefer 4 or more hours of sun.

Some herbs to try might be American pennyroyal (I think there's another mint that does well in shade, too, but can't think of it right now), sweet grass, chervil, and bee balm (I think?).
My purple coneflowers (echinacea) were supposed to be planted in full sun, but they're hardy little flowers that are doing fine in a little less than 4 hours of sun here, so you might try those, too.
HTH! Best of luck!
--Amy