Tomato paste has such intense tomato flavor, its richness and depth complement many foods. Often I would open a can, leave it in the refrigerator, only to discover it much later, dried up and useless. What a shame.
I have discovered that if I tightly wrap the portion I do not use in plastic wrap, and freeze it, it is a simple thing to cut off what I need and add it to my dish. The remainder is then kept frozen. Of course, now there is tomato paste in a tube... wonderful!
I stir a small amount in with fresh, hot pasta along with a little butter until it is all mixed in and then add fresh herbs, salt and pepper, sprinkle with Romano cheese. Sometimes I add sun-dried tomatoes. Zesty!
When cooking green beans, I take a can of green beans, drained, put into a skillet with sizzling butter in which I have cooked onions and garlic until limp, then as it is all coming to a sizzle, add a tablespoon or two of tomato paste. Season with salt, pepper, seasoning salt, a drizzle of lemon, turn down the heat, cover and simmer for about two minutes. This is really tasty.
I use tomato paste in beef dishes, especially left-over meats. I cut them up, cook with onions and garlic in a little butter, add the tomato paste, and simmer with a little water. This sauces the meat, warms it through, and gives it an extra kick. Add pepper flakes if you like it spicy.
One of the best ways to use tomato paste is to mix in with cubed potatoes, seasonings and cut up chicken thighs... I cook the chicken parts first, then add the potatoes and cook all of it together,covered, after adding the tomato paste until the potatoes are fork tender. It's flavorful and so delicious.
I am certain you must have uses for this great sauce besides making pasta. It has a sweet, condensed flavor, lifting anything it is cooked with. Go on, experiment!