Ok, most auto parts stores will check for codes for free, I would recommend that even if the check engine light is not on, but it sounds like a fuel system issue as you mentioned. If it only has an issue on restarting when hot I would suspect a bad check valve in the fuel pump allowing the fuel rail to drop pressure and run rough as it is starving for fuel until it builds up pressure. You can check this with a fuel pressue gauge.With the key on engine offyou should have 30 psi of pressure and hold steady, if it slowly bleeds off, I would suspect a bad chck valve in the fuel pump. Make sure you have a newer fuel filter as it could be restricted as well.Also you can try cycling the key to the on position a few times to build pressure back up and see if it still has a rough restart. if it does then a fuel injector could be leaking down but I dont think thats the case since it runs ok first start of the day. If an injector was leaking down into the cylinder it would run rough on cold start as well.