Sorry I don't recognise the problem, one of the biggest advantages of a Rohloff for me is the ability to change down quickly and reliably on a hill.
As has already been said it should be changing with ease in under a second, if you don't have the momentum to carry you through that, then you've left it too late, but that would be even more so on a derailleur bike.
When you do change, having your predominant leg at about 1 o'clock, as you would from a standing start, might help. As might putting maximum effort into the revolution before. If I find my cadence falling below a sustainable level, I'll sometimes change up a gear, stand and honk to increase speed, sit and immediately drop a couple of gears. But ideally I wouldn't find myself in that position and I certainly wouldn't be confident could do that on a derailleur bike.
Maybe it's just a question of familiarity, the way I ride is slightly different, but IMO the advantages are all on the Rohloff's side, I tend to forget this till I go back to a derailleur bike.
General hill climbing "Tricks" though I'm not sure you're asking for them:
Count your pedal strokes and put extra effort into the third, one, two, THREE, one, two, THREE, one, two, THREE ... When that gets tiring, make it the fifth. That one extra thrust will help carry you through the next.
If you have stability issues, particularly the front wheel lifting, bend your elbows 90 degrees. It's incredibly difficult to lift the front wheel, even loaded on a hill, unless you're pulling up on the bars. Bending you elbows means your always pulling back instead. I know there are those who disagree, I used to be one of them!