I've only done a handful of them, mostly with really fine wire. I always use my twisted messes though... ...it's got plenty of clearance for those post holes, so I guess I've never thought to worry about snags.
I tend to wrap the first and last few twists completely vertical - as in, a clump of the thinner wire wrapped all up over itself on both ends. It's just something I do out of habit from doing claptons most of the time.
I don't think it will matter too much if the thinner helixed wire pops up a bit at the posts or doesn't make contact at the connection sites. I don't think it's electrically active, as per clapton theory (the cores are the lowest resistance path by an exponential margin, so the current will travel from the highest resistance [helixed leads] path to the lowest one [base twists] right at the point first point in the circuit where they all make contact [ usually the connection site.])
...or at least that's what I'm observing with my helixed claptons. Might be a different story with standard helixes. It would depend on the tightness of the twists and the resistance differential between core and helix.
If the latter is what you're working with and you find that the thinner wire does need to make contact, then you could try bending the ends of the leads and pinching them tight with pliers. That's what I often do with claptons to keep the outer wire from sliding off of the core. Should be even easier with helix wire. Width might be an issue when getting them through the post holes, though.
Or maybe you could just grab the wire real close to the chuck (like 5 mm) and give it a few quick turbo twists to lock it all in. Repeat with both ends to secure the outer wire. It's a potential option that's less bulky than the bend n' pinch... ...if you manage to do it without snapping the wire.
