I've asked this question to a couple of physicists a few days ago. Great minds think alike, huh?
First, bear in mind that Hawking radiation is only hypothetical. It is not theory. If we trust that hypothesis, this is what we can get.
In general relativity, black holes can be described through a number of approximations. For example, the Schwarzschild solution for a black hole describes it as an eternal object — not something that exists for some times, and doesn't exist for others. According to this solution, the event horizon must have always existed, and must remain existing eternally.
Schwarzschild black holes approximate black holes very accurately, but as you can tell, they fail to explain how a black hole can form, and (assuming Hawking radiation is real) they don't explain how one could eventually evaporate.
Of course, that solution won't help us. I've kept looking for one that accurately describes an evaporating, creatable black hole, but I've found nothing. The conclusion I've come to, along with those I've asked, is that our question has a major problem: Hawking radiation is explained via quantum field theory.
Thus, you can't simply use a GR solution for a black hole; you'd need some unholy mixture of quantum field theory and general relativity (keep in mind that both GR and QFT are incompatible in many situations).
In the end, it all comes down to how little we really know about black holes. It's not really possible to determine which solution is the best, and our inability to reconcile QFT with GR poses a big problem. The best answer I could give is "nobody really knows what would happen if you kept approaching a black hole".
We don't know if we would reach the event horizon, we don't know if the black hole will evaporate. We simply don't understand them well enough to know what solution would work, or how we would put QFT into it. If we somehow managed to find an approximation that properly combines GR and QFT, I assume (but don't quote me on this) the situation you described would be possible.
If it is possible, by the way, then we could confidently say that a black hole of any size could rip you apart through tidal forces. Tidal forces become weaker as the black hole's size increases, so one would assume a large enough black hole wouldn't rip you apart.
However, if we take Hawking radiation into account, and if your proposed scenario were indeed correct, the black hole would shrink as it evaporates. Since it would get smaller at a faster rate as we approach the event horizon, it would soon be small enough to rip us apart.