Is it possible to go out after going to the Metaverse?
Luckily, there is a way to cure exhaustion in Persona 5. After reaching the maximum rank for the Temperance Confidant, you unlock the ability to get a massage after traversing the Metaverse during the day, which allows you to go out that night.
You unlock the Temperance arcana Confidant on 5/24 and you will need at least rank 2 Guts to start the Social Link.
It's worth noting that while this Confidant is only available to spend time with on Friday and Saturday nights, you are able to get a massage on any night.