To put away your wand while exploring in Hogwarts Legacy, you simply have to stop moving for five seconds.
After being idle for that long, your character will automatically place their wand back into their pocket and will no longer carry it about with them until it is used again.
It’s worth noting that if you cast a spell and then wait, your character won’t put away their wand. So, after casting a spell, move around a little and then wait five seconds.
Additionally, the type of wand you’ve chosen or the spell you use doesn’t seem to have an effect. Both combat and exploration-based spells cause you to have your wand out until you stop moving.
So, you could use any of Hogwarts Legacy’s many spells, and your character will still keep their wand out indefinitely.
However, this is not the case for books, which your character will hold after finding Field Guide Pages. You will automatically put the book away even while moving, so you don’t need to worry about waiting around for that.

Can You Manually Put Your Wand Away in Hogwarts Legacy?
No, there is currently no way to manually put away your wand. No button prompts will force your character to holster their wand.
This issue is yet to be addressed by Avalanche Software, the developer of Hogwarts Legacy. However, as it is a highly requested feature, it will likely be added later.
For now, players must wait five seconds to allow their character to put their wand away. While it is inconvenient, it is the only way for the time being.
Of course, in the meantime, if you’re going to be seeing a lot of your wand, you may as well complete the many side quests in Hogwarts Legacy that grant you new Wand Handles.