What's the best approach for extending stopovers on Greyhound? Do I need to ask to keep the first leg's ticket?

It’s been handled differently on different trips I’ve taken in the past and I can’t get a clear answer from the Greyhound phone line about the right way to do it.

By extending a stopover I mean for example I have a long trip from City A to City C with a transfer scheduled in City B. On my ticket it says we’ll spend one hour in City B, but I want to spend 12 hours in City B because it has a cool museum or something.

Sometimes it’s been fine to just get on any subsequent departure from City B; sometimes they want me to go to the ticket counter and switch my ticket; but one time they warned me I could only do that if I kept my previous leg’s ticket (the part they would normally take from me in City A as I boarded the bus for the leg from City A to City B).

Thank you for any experiences/advice!