It was explained to me when I first worked for a carrier with CASS that the jumpseat belongs to the captain, whether it's the physical one in the cockpit or the figurative open cabin seat. Consequently, you always ask when using something belonging to another. For me it's:
"Hi, I'm xx with xxx airline. Can I hitch a ride with you today? They have space in the back." Or, if necessary, "They said it's full, so can I share your space up here?"
If I didn't ask, I would assume that I would be asked to deplane when the head count is delivered.
One time I got stung riding on Southwest from Tulsa to Detroit. The flight had a stop in St. Louis, but continued on to Detroit as the same flight number. It was very convenient for me to not need to change airplanes. I asked permission as usual in Tulsa. There was room in the cabin. In St. Louis I failed to observe there was a crew change, and then the airplane filled up. There was a PA asking for the jumpseater to come forward. I did, and the new captain pulled me onto the jet bridge to give me what for. It was kind of embarrassing. The outgoing crew didn't brief the incoming ones or leave the paper ticket that Southwest has one give to the captain. If I would have noticed the crew change, I would have renewed the permission. Eventually things got sorted out, and I was allowed to ride in the cockpit from STL to DTW. I thought that second leg captain could have handled things a little more gently. For awhile I thought I'd be left without even getting to go back and retrieve my carry-on bag.