Intermittent starting issues after being driven?

This one has been driving me nuts. It's a 2003 like mentioned, pretty decent shape, new good battery, starter is pretty new, no real issues, also we are in a hot and coastal climate that has a lot of salt in the air. If I start it after not driving it for some hours, she starts like a champ; easy and butter on first try. Runs well, regular rpms, no rough idling, no bad symptoms. Here's where it gets funny; I've found that after I drive it for a little bit (typically over 15mins) and turn it off/park. It will start up right away just like the first start of the day - easy. I've noticed that if I leave it for a few minutes (like turning it off and running into a grocery store), that most of the time; it won't start right back up. The battery still has good voltage, the starter tries, but it doesn't turn over for some time (sometimes up to 30-40 minutes of sitting there and waiting), but then it comes back to life and starts. The only other thing I can add is that I noticed when trying to turn it over, sometimes it makes some strange noises in the exhaust like a backfire and sometimes there is a bit of a noise in the engine which kind of sounds like a thud. Any help is appreciated. Thanks!