I'm not sure where you got the idea that pinching would encourage new vining stems. It doesn't. It might encourage bushiness but not more stems (which is what I think you mean by "branches").

The additional stems/branches will originate only from the root crown. You encourage these by A) planting the root crown deeply (4-6") in a generously sized and well amended planting hole and B) hard-pruning the vine in late winter/early spring for at least 2 growing seasons.

Also, fertilizing with alfalfa meal or alfalfa tea will also stimulate additional stem development. Alfalfa contains a natural plant growth hormone that promotes this type of growth. But it is getting a bit late to do too much more fertilizing. And don't expect it to happen very rapidly. If you follow these practices, I wouldn't expect see any changes until next season's growth begins.

Ah, very helpful! I must have read that pinching would encourage "fullness," and then I just assumed that meant new stems. But what you're saying makes sense. I'd like both fulness AND new stems, so i will try hard pruning in early spring and alfalfa next season. Thank you so much!