A method beats inspiration. For every position, ask these in order.
1. List the checks
In a tsume only checks count, so the candidate moves are few: every drop that gives check, every piece move that gives check. Write them down (in your head). A 3手詰 rarely has more than eight or ten.
2. Count the escape squares
For each check, look at every square next to the king. Which does your checking piece cover? Which are covered by others? Which does the king's own army block? The mate is the check that leaves no escape — or leaves escapes you can close next move.
3. The king's head
The square directly in front of the king is where most mates happen — the gold on the head above all. A gold there, protected, covers five of the king's eight neighbours.
4. Sacrifice to open or to draw in
If no check mates, find one that forces the king where you want it — often by sacrificing a piece the king must take. You'll get pieces back: whatever the defender captures, you can't use, but whatever you had in hand stays.
5. Use your hand last
Pieces in hand can go anywhere, so they're the most flexible thing you have. Moves on the board first often keep the hand pieces for the mate itself.
Try it: a 3手詰
Try it: Mate in three (3手詰). Check with a drop that drives the king toward your knight.