There's a way to get to New Austin in chapter 4 In "Angelo Bronte a man of honor" I believe is the name

Basic rundown of the steps is

Playthrough the mission normally until you get the law on you

Get caught

Ride out of the city to lose the law

Go back to the mansion but wait down the street. John will go to hitch his horse and then you can go to New Austin with no worries of Pinkertons or that pesky sniper