Identifying wires in strange configuration
You've already identified switched-hot.  The old lamp would have needed it, of course. 

The old lamp would have also needed neutral, and you've identified one of those.  

Lamps generally do not want/need always-hot, but if there are 2 or more cables in a lamp box, fair chance always-hot is in there.  Since the lamp doesn't tap it, it's most likely that undisturbed 2-wire bundle in the back.  Typically you have two always-hot's in a lamp box when always-hot is also being carried onward to other points of service - outlets and the like. 

Generally always-hots and always-neutrals show up in pairs.  E.G. AH+N from supply and AH+N onward.  You have two always-hots, so you might expect two neutrals. (All neutrals are always).   You don't have 2 neutrals identified (yet). 



So what's left?  A wire that reads "hot".  And this wire was disconnected as part of removing the old light, which makes it very likely a switched-hot or neutral.

When you run through the possibilities -- a multi-wire branch circuit's second "hot" is out; that wouldn't be connected to the lamp.  

Switched-hot onward to another lamp would leave this box awfully starved for neutrals - you need a neutral supply, a neutral for onward power, and supposedly a neutral for this other lamp!   And you don't have another lamp. 

Now, when an onward circuit is partially disconnected -- obviously if you disconnect the hot, the hot reads 0V.  But if you disconnect the neutral, current moves through the loads and tries to return through the neutral, and it can't.  So it lifts the neutral up to line voltage.  Or phantom voltage lifts it somewhat.    

That is exactly what we are seeing.  And it covers the neutral we'd expect as the two neutrals for the two always-hots.     So, join the mystery wire to the known neutral and the lamp; you should be all set.