I had this problem with my old bike last year. I finally determined it was my left pedal/cleat that was squeaking and if I changed the angle of my left heel slightly the squeak would go away, until I forgot about maintaining that angle and then it would come back.
Replaced the bike this spring and swapped the pedals over to the new bike. After a couple of rides my squeak was back. Like you it took a few miles of riding before it started.
I finally got tired of it and decided to figure out if it was the pedal or the cleat. I swapped the cleats from right to left and I haven't squeaked in a couple of hundred miles now.
When I had both of the cleats off I examined them closely and although they both showed some minor wear there wasn't anything obviously more worn on the left than the right.
For now I'm happy, no squeaks and if and when it does come back I'll probably just see about getting a new set of cleats.