The most obvious answer is nostalgia... But I think we can take the analysis a bit further than that.
It was just his will to know whether he's still worthy or not, which he was.
Thor has a "special and intimate relationship with this hammer". He's wielded it in every battle, and at the end of the first Thor movie, Mjölnir coming back to him meant he was worthy again. The last time he saw Mjölnir, it got destroyed from his hidden Hellish sister, and it took him a long time to understand he wasn't the "God of hammers", and could operate without it. Still, you can't beat the nostalgia.
By Endgame, Thor is in a bad place. As Paul developed in this excellent answer, Thor lost pretty much everything by Ragnarok: his mother, his father, his hammer, his freedom, his cape, his hair, his eye. By Infinity War, it's worse: his brother died, as did half of his people and the rest of the world, that as a superhero he had a moral duty to protect. By Endgame, you'd think it couldn't be worse, but it can: Thanos destroyed the Stones, nobody can do anything about what happened. There's just no point anymore, and Thor must be pretty depressed.
Make all the "fat jokes" you want (or better yet: don't), Thor knows he's not "fine". So when the gang is back together again, with a purpose, hope, and he is given the chance to tell his mother goodbye (sort of) and he discovers he's still worthy, when he had doubts he could (despite the after-fact "I knew it!")? Immediately followed by troops charging with Norse artillery to thrash his raccoon pal and him? Thor won't stop to think about it. He's whole again, and Mjölnir is part of that.
Thanos was long dead and there was no war to be fought at that point of time.
Irrelevant; Thor is now "back in action", and action means the hammer, even if the plan is only to unsnap and hope everything works out fine.
Plus, heroes always prepare for war, right? That's what they do, especially Thor, who is almost born for fight.
What were his plans after the snap? To keep both the weapons in his inventory, while the Thor in the alternate timeline pulls his hair out while searching for his dear Mjölnir?
Thor and plans, not two words that go along well. MCU Thor, even in his best day, was never that much of a thinker/planner, and the time travel consequences are probably not reaching him. For all he knows, as they have the Stones, if anyone says something he'll be able to return Mjölnir in no time (pun intended).