With some of my students, particularly the ones who are learning BG/OT/fiddle tune-type material, I've sort of incorporated John Moore's idea of breaking the tunes into "words" (apologies to John Moore if my explanations are poor).
If you think of the whole A part, for example, as a sentence, that sentence can be broken down into its component words, or melodic fragments (usually but not always corresponding to measures). I have my students drill these fragments individually so that they're completely ingrained.
Say there are 8 measures. Say it! Just kidding. I'll have them play the first 2 measures, for example, as usual and then alter the 3rd measure in some way, coming back to the familiar melodic fragment in the 4th. This teaches a few things: how to alter a phrase in bite-size pieces; how to get back into the tune without waiting for the next A or B part to happen (something I see lots of fiddle tune players have trouble with). We'll vary which measures we want to change, eventually changing several in a row and trying to make our explorations make melodic and harmonic sense (not just running scales). At some point in the above process, we've begun improvising! I hope all that makes sense.
Another thing I find extremely helpful is singing what I'd like to play. It seems that most of us can improvise very well with our voices so the challenge seems not to lie in the ability (or lack thereof) to improvise but rather in the ability to translate the ideas from brain to hands. I'll sing an idea, learn it on the fingerboard, then sing it while I play it. Eventually (hopefully) the connection will grow stronger and I can "sing" those ideas with my fingers instead.