Common ingredient to thicken the soup
There are lots of potential thickeners, but you often need to select the one that works best with your given need (temperature, if it has dairy, resulting mouthfeel, etc).

In your case, you're already using rice, so you may want to stick with a starch -- corn starch, potato starch, tapioca, etc.  For these, you add a bit to cold liquid, mix it well, add it to the soup and heat it up.  As it approaches boiling, it'll thicken.

You can use flour (wheat, rice, yam, etc.), but in the case of wheat flour and possibly the others, you may have to deal with a raw taste.  You fix this by cooking the flour with butter or oil into a roux, and then adding that, and heating it up to a near boil.

You can also use pureed vegetables.  Cook them in the soup, then put it through a blender so the vegetables add body to the soup.