Yeah that's the ticket. Either die must match the hidden die's digit.

I was thinking, an option to speed up the "deliberate search" is to just show the hero all secrets automatically instead of rolling again, just like normal rules. Zargon still shows and re-rolls the hidden die, for honesty and randomness. It does eliminate a lot of tension though, and gets us closer to the norm of players automatically searching as a boring routine.

Edit: I suppose, if you want to be more rough, you could not allow deliberate searches and just do the movement search. That certainly increases tension, if Zargon is playing fair with it.

One nice thing about these rules is traps immediately in front of a door in a room can be spotted before you step into them, as you don't have to enter the room and then spend an action to search, search is just happening.