1998 Ford F-150 starter

I just replaced the starter as the starter would not engage (bad solenoid). Put the new starter on in my heated garage. Started it up, everything worked fine. Backed it out of the garage and parked it in the cold (subzero degrees F) overnight. Went to start vehicle in the morning and the vehicle started but would not disengage, even when the iginition was turned off so had to disconnect the batter. Went back to parts store and swapped the starter for a new one and replaced it again. Same story with second starter. Everything seems to work fine when the vehicle is started in the warm garage, but not in the cold. Any ideas?