This video includes a complete trip. A completely new way (without using any other missions) and the best route (most timesaving). You may want to watch till the end if you love this horse or welcome the challenge. Here are the timestamps and watch based on your need:

00:00 some ways to die (of course a lot more, lol)
00:33 foreword + the route in 4 phases
01:15 Phase1
02:55 Phase2
03:20 Phase3
05:20 Phase4
08:15 the Exit out of New Austin + the horse is yours

If there is need, I will make a Q&A video on this task. Been on it for some time and died and failed hundreds of times and hundreds of ways, lol, recorded on camera. I think I am able to answer all the questions related to this.