I have had two elite fishing poles (underlight angler) disappear when reequipping my weapons. I was
able to recover both poles from the twisting nether by buying a vendor pole and equipping it. That
solved the big problem.
That said I am only getting the base +5 bonus to fishing from wearing Nat's Hat. I am not getting the
+10 from the lure, the +6 from the line (1 pole) , the +15 from the Underlight Angler when it was
equipped and the +5 from my glove enchant.