First of all, you must understand that genetics play a MAJOR factor.
Someone taller than me would find vertical pole hanging more difficult, while someone shorter than me would find it easier. Also, someone who is top-heavy would find it easier than someone who is bottom-heavy (if both were about the same height). It's just a plain fact of physics.
Having said that, let's get to your question concerning the training method that would be the most effective to help you reach your goal to vertically hang from a pole. But first, let me explain the stress my body experiences while hanging sideways. While hanging sideways, the side of my body that faces the sky feels the stress along my lat, bicep, forearm (grip), and abs (very heavy stress in the ab area). The side of my body that faces the ground feels the stress in the tricep and shoulder. You may find that you can hang more comfortably and longer on one side than the other. As you can see from my website, I'm hanging on the same side in both pictures. That is, my preference is to hang sideways with my left arm lower than my right arm. However, I can hang on both sides.
Now to the training. Before my three shoulder operations, I competed in ADFPA powerlifting (drug free all my life). So my training method consisted heavily of squats, deadlifts, and benching. In other words, you need to get STRONG! I trained on alternate days (never two days in a row). Here's a routine that I have recommended to people and they are making huge increases in their strength (HIT method):
ROUTINE I:
Squat or Deadlift: 1 set (alternate each workout)
Calves: 2 sets
Shins: 2 sets
Leg Ext: 1 set
Leg Curls: 1 set
Hyper Extensions or SLDL (alternate): 1 set
Abs: 1 set
Neck Bridging (front, back, side-to-side): 1 set each
ROUTINE II:
Flat Bench: 1 set
Incl. cable flies: 1 set
Under-arm grip pullups or regular pullups (alternate): 1 set
Rowing: 1 set
Seated DB press: 1 set
Lateral or Front DB raises (alternate): 1 set
Bentover DB raises: 1 set
Internal rotations: 1 x 15
External rotations: 1 x 15
Shrugs: 1 set
Dips or C.G. press or Lying tricep exts (alternate): 1 set
Barbell curls or Lying incline DB curls (alternate): 1 set
Wrist curls: 1 set
Rev. DB wrist curls: 1 set
Spring pulls (grip): 1 set
I did not indicate the number of reps in the above routines (except for the internal/external shoulder rotations). I cycle my reps. I begin each new cycle at 20 reps and cycle down to 3 reps over a period of ten weeks. At the end of the cycle (the 11th week), I attempt a 1-RM on the Squat, Deadlift, and Bench.
Then just keep repeating the above. Again, never two days in a row. I also do 15-20 minutes of aerobics every morning (bike or run in my case. I usually alternate the two).
You may need to experiment with the number of recovery days needed between workouts. I wouldn't add more sets. Adding more sets indicates (to me) that you are not training hard enough. There are exceptions to this rule. Some people have injuries that prevent them from training "all-out". Doing so would be too painful and dangerous. Consequently, they may find it beneficial to reduce the intensity and do a few more less intensive sets.
Lastly, remember what I said about physics. The above routine will get you very strong. But if you are considered "tall" and/or "bottom-heavy", you may never be able to hang vertically. Also, the above routine works wonders for people I train. It may or may not work for you. If you do choose to give this routine a try, please try to stick with it for at least three months.