Hi - I replaced my [defintely bad] starter last week and hoped somehow a problem I've been having for awhile would go away. It did not!

This problem started in September. It usually happens in the morning after the car had been running about 5 minutes (at < 30mph) and then I turn it off, then return to start it again about 5 minutes after that. The lights go on the dash but nothing happens. Not a peep from anywhere! If I continue to turn the key, maybe once every 5 seconds, it will eventually start, usually within another 5 minutes.

The starter replacement was my very first car repair other than changing flat tires. I have not learned much about diagnosing things like this yet. Any help given will be really appreciated! Thank you so much!! Marcia