DO NOT!!!!! tell the mechanics what to look for. If you do that, they will concentrate on what YOU told them and may miss the REAL problem. Instead, just tell them that it's using too much oil and let THEM figure out why. That's what they get paid for.
Could be something as simple as a stuck PCV valve, sucking oil into the engine and burning it when it shouldn't be.
I am not familiar with your car, so this is just a suggestion. I have seen PCV valves cause oil to leak out because of over pressuring the engine, but it can also stay open and allow oil to be sucked into the intake, thereby burning it.
Other then that, it could be internal, bad rings, allowing oil bypass.
But as JetDoc says, do not tell your mechanic what you think, let HIM tell you.
Here's a perfect example why using the proper maintenance and quality engine lubricant, ie. synthetic. When petroleum based oil is used and then not changed as often as it should be only wears out the engine prematurely. I'm guessing the engine is worn out and no magic oil will fix this for you. If I were you'd use an SAE that quietens it down and then sell it to a red neck from the oil patch who doesn't care much about how much oil it burns.
Unfortunately, these Corolla engines are known for burning oil. Switching to 20W-50 was a really bad idea. You actually create more engine wear with that oil. It doesn't flow as fast on cold starts, generates more friction, runs hotter, and shears faster. Try high mileage oil at the recommended viscosity instead. If has film strengthening addivtives to seal better without causing extra wear.
JetDoc is right on the money about telling a mechanic what's wrong. Unless you regularly have a pool of oil under your car you have a serious engine problem. You are another candidate for the clean it up and sell it asap therapy. Go ahead and lie a little about the oil issue -like "oh yeah it does use a little oil but oil's cheap" It's OK to lie when selling a car. This tradition was established by the used car salesman's union in 1946.
There is a simple check you could do yourself before taking it to a mechanic, just to have a first idea what is going on. If you see smoke from the exhaust when accelerating( a continues cloud of smoke), then it can be piston rings or PCV. When you let it idle for a minute or so and then push the accelerator and then a short plum of smoke comes out, its most probably the valve stem seals.
Tell him nothing let he or she give you they're opinon
I am currently in a 2003 corolla (118,000 miles on engine) that is eating about a quart of oil every 700 miles. I changed to a much thicker oil grade (20w-50). This did not really slow the rate of consumption by much at all. There is certainly not a leak, it is burning the oil out. Is there any particular thing that I should ask the mechanics to look at? Ideally, I don't want to have to do a major overhaul of the engine.
My question is: What should I have the mechanics check for? And is it possible (or likely at all?) that a smaller fix could help solve this issue?
Thank you!