You have another Battle immediately.
On page 5 of the game rules:
If at any time during the game the board is filled up and no single space is left to place a tile on, a Battle commences (see BATTLES section below).
Even if the Battle results in a stalemate, the game will eventually end. As soon as the Battle ends, so does the active player's turn:
Looking through the Neuroshima Hex 3.0 rules (pdf), I see the following:
So the extra hexes are for when there are five to six people playing. The rules suggest that if there are four people playing, you might decide to use the expanded board as well.
This rule book doesn't mention scenarios. However, I looked up one of the expansions: Neuroshima Hex: ...
Yes, it does. A battle will always end the turn. From the 3.0 rules:
The player who starts a Battle (by playing Battle tile or filling up the board)
finishes his turn, regardless of the number of tiles he has left. As soon as the
Battle commences, no other tiles can be used or even discarded.
It's also the way it is implemented in the mobile app (I'...
Discarded tiles are out for the remainder of the game. While the rules don't specify this, they never allow for reshuffling of the draw pile. This would also force you to maintain separate discards for your defeated tiles and your discarded-from-hand tiles, which is an unnecessary and unspecified complication.
As further proof, the behavior in the ...
You can make some hexagonal tuckboxes, check out these ones from BGG.
You may need to tweak them a bit to deal with your custom tiles, but Inkscape does a pretty decent job of editing these types of PDF's (YMMV)