We also used eyehook latches for rooms where SMNH is supposed to stay out. (For the keep-in/keep-out problem, could they be installed at the 5-year-old's height? Though what to do once the 5-yo's in...)
I haven't tried the squeezable door things but have seen them and was going to suggest them as well.
What if you take one or two areas where he wasn't allowed before (was he allowed to wander into his room during the day?) and make them child-safe for him? That way you can play up the "look, exciting new area for you!" and let him go there at will, and he might be less inclined to try to beat whatever new thing (hooks, door squeezer) you put on the rooms you really don't want him to enter.
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I haven't tried the squeezable door things but have seen them and was going to suggest them as well.
What if you take one or two areas where he wasn't allowed before (was he allowed to wander into his room during the day?) and make them child-safe for him? That way you can play up the "look, exciting new area for you!" and let him go there at will, and he might be less inclined to try to beat whatever new thing (hooks, door squeezer) you put on the rooms you really don't want him to enter.