(We are out of town for the holiday weekend, so no recipes today. I hope you find this information helpful.)
One of my favorite kitchen tricks is to keep cooked rice in my freezer. Whenever I make fresh rice, I make a double batch. For us, this means four cups of water and two cups of rice. I serve it all for dinner and then put the leftovers in a zip-top bag in the freezer. (I am moving to glass containers but the move is slow.) It is best if the rice is still a little warm when you put it in the freezer as that helps to capture liquid in the container. If it looks dry when you take it out of the freezer, add a smidge of water before reheating. (Like a teaspoon smidge, not a cup smidge.)
Whenever we need a quick side, or I want to sneak rice into an unsuspecting recipe, I grab a bag out of the freezer. If I've planned ahead, I can let it defrost in the refrigerator or on the counter. If I'm in a hurry, I can open the bag top a little bit and put the whole thing on the microwave. I don't like microwaving plastic, but it turns out perfectly. The rice has enough water in the ice crystals to make steam inside the bag and reheat well, and the little big of opening at the top keeps the whole thing from exploding.
If I am feeling guilty about microwaving plastic, I pull it out of the bag and put it in a bowl. It microwaves great but it does dirty an additional dish and I don't get to marvel at how amazing it is in the bag. Whether you need to cover it depends on a couple of things: the kind of rice and how much moisture it retained, what you are planning to do with the rice, and your own preferences. You can cover using plastic wrap or a plate slightly offset to allow a little bit of steam to escape.
If you are concerned about microwaving plastic, but glass sounds like a pain or expense, another option is using parchment paper or waxed paper to wrap your rice. You'll have to put the wrapped rice in a plastic bag to keep the moisture in, but you can microwave it in just the paper and you can reuse the bag forever. Rice reheated in the paper might not be quite as fluffy as microwaved in a plastic bag but it will certainly do, especially if you are adding it to a recipe.
Shorter grain rices (arborio, sushi, medium grain) freeze and reheat better than longer grain rices (jasmine, basmati, Carolina). This seems to be because the shorter grain rices have more moisture, which makes them freeze better and then they are able to use that moisture when reheating.
There is a toxin (Aflatoxin) that can grow in cooked rice if left at room temperatures too long. Get that rice into the freezer and be safe!