Starter Solenoid

I have a problem where the starter is not disengaging from the flywheel for a second or two after I release the key.
Since the starter was original, with 120k miles on it, I replaced it first.
Still experiencing the same problem, I checked the solenoid mounted on the driver's fender under the hood.
I noticed that it's new but definitely an aftermarket one.
I seem to remember having a similar experience with an aftermarket solenoid years ago.
I've ordered an OEM Motorcraft solenoid.
Has anyone else ever experienced this issue?