I'm studying in-person in the US and graduating this month. I have recently received an employment authorisation document (EAD) for optional practical training (OPT), valid until May 2022. Also, I have been approved ("... we approved your Form I-129, Petition for a Nonimmigrant Worker ...") for the H-1B visa and I understand that I will automatically transition to H-1 status on 01 October (source) - so far, so good, and a fairly common situation to be in.
I have had a hard time, though, finding an answer to an obvious question anywhere - what happens on my first trip abroad after 01 October? I have only an F-1 visa in my passport at the moment, and instinctively I think that when I return to the US there will need to be a new visa in my passport reflecting my new H-1B status. Is my instinct incorrect, and am I able to reenter the US without a corresponding visa in my passport as a result of this change of status? Or is my instinct correct, forcing me to plan a lengthy interaction with a US Consulate on my very first trip abroad where I would have to book an appointment, hand them my passport, then wait in that country until I get the passport back with my visa?
I have used the term "visa in my passport" wherever applicable to avoid the confusing terms "stamp"/"stamping". Still, I am happy to edit my answer with additional details as required.