Unfortunately you have it back to front. The other way around a 60 day TR visa would have been cool with a 30 day exempt on your return for the 5 days. The way you're doing it you could either arrive visa exempt, move to your next country and apply for a 60TR there if you have the time or arrive visa exempt, visit your other destinations, return on visa exempt and apply for a extension at your nearest immigration office
Im doing this next February. Going skiing in Hokkaido but am going direct to Japan from Australia and then to Thailand so I'm just getting my visa in Oz. I think to do the same in the US is your best option
I'd do as Charlie mentioned. If it's a 77 day trip and you are visiting other countries, no need to screw around with a visa. But definitely take warning about the airline asking for an exit flight in 30 days. That is very possible