Extending power from 3 way switch with power coming into the fixture
Your problem: too many hots, not enough neutrals, because the prior installer didn't know how to pigtail
Your problem is that in the fixture box, you somehow have two wires carrying always-hot power down to the switch but nothing carrying neutral to the switch location.  Atop being a violation of 404.2(C) in the current NEC, this is what is stymieing your plans to tap power for the hallway circuit at the switch box, as you need two to tango: a hot to get power from the utility, and a neutral to return the &quot;spent&quot; electricity back to the utility (they want their electrons back, so to speak :).
Anyway, what you need to do is get into the light fixture's box and move the white wire in the 14/3 over from being nutted in with the always-hots to being nutted in with the existing neutrals and remove any tape flags or bands from said white wire while you are in there.  Once that's done, you can button the light fixture back up and move onto the first switch box.
There, we take the tape off the white  wire from the 14/3 to the light fixture box, remove it from the switch common screw, and then nut it to the white wire in the 14/2 exiting that box.  From there, we take a black or red 14AWG pigtail (if you don't have any suitable wire laying around or shuckable scraps of NM handy for that matter, simply get some THHN of the appropriate gauge and color and use that) and nut it in with the junction of the red and black wires in the switch box, then land the other end of the pigtail on the black common screw on the switch that we just vacated.  Once that's done, you should be able to button the switch back up, turn the power on, and enjoy having everything work again!