Engine stutters when started warm

I have a 1999 Corolla with about 60,000 miles on it. The engine starts fine when it is cold. But if I start the car the when the engine is warm, the engine "stutters" for a split second and results in a very short "shuddering" movement. Once started, the car runs fine. This has been happening for about six weeks.

My old mechanic just moved out of the area and I just started going to a new garage. This mechanic said a tune up and a fuel system service are needed. I can understand a tune up is likely needed, but I cannot see how a fuel system service is necessary at this point. I am inclined to do the tune up first and assess the results. What do you think? I will be pleased if you can see less expansive ways to get it fixed!