Presumably the answer is different for different films, and when we say that a film pushes well (in terms of recoverable shadow detail), we should be saying that a comparatively short exposure gets you shadow detail in the latent image that isn't activated under normal development; that is, it doesn't take too much exposure to get the film "activated enough" that it doesn't develop to completion under a normal regime. (In other words, if you looked at the "intrinsic" characteristic curve of the film---exposure vs. activated grains, without taking any development into account---the toe would be steep.)
Futzing around with Tri-X has led me to believe that it has that kind of "pushes well" property, because sufficiently extreme development (e.g., Donald Qualls's "Super Soup" concoction) produces shadow detail that seemingly "shouldn't" be there. See the attachment for an example---that's TX400, metered fairly carefully at EI 3200, and I don't look at it and see "dead vacant shadows". Obviously there's some loss of detail, but, for instance, you can see the contours of the drawers in the cabinet at right, and the grain in the table under the shadows of the subjects' arms. I haven't done spot metering and densitometry to validate this impression, but experientially, I think there's quite a lot more detail being dragged out of the shadows here than in normal development.
I'd agree certainly with all that. I've seen some pretty impressive results claimed for Rodinal and stand or semi-stand development, but not being a member of the cult myself, I have no real way to judge how typical that is.The OP was asking about using a developer that wouldn't help much at all. HC-110 and Rodinal are not the best for retaining shadow detail in a push-process situation. X-tol, T-Max developer, Acufine, Diafine, some of the Ilford developers will do a better job.
I did once try an experiment in really long semi-stand development in HC-110, with TX400 exposed at 3200, and while the results weren't in any technical sense *good*, they were unquestionably *interesting*. No great increase in shadow detail, but lots of contrast and completely absurd amounts of grain---some people have said they really liked the images, but most just get a funny look on their faces and try to avoid saying anything insulting.