There's nothing in the Pushing section that says you can't push a roll you only have one die for, is that somewhere else?
(You can't push if you rolled a mishap, and you can't push passive rolls, and you can't re-roll a 1 when pushing, but if I have Agl B and no Mobility, I can push a Mobility roll if I fail and don't roll a 1)
As you said, you can push any skill roll as long as the skill roll isn't a mishap (and yeah, you can't re-roll 1s).
edit: When it comes to mishap, I think people are over-thinking it. To have skill level D modified by -1 will result in -1 step so you now have no skill levels for this particular skill roll.