What should an explorer's "Day Job" be?

There isn't really one, or any. It can vary because the Explorers mood can vary.

For the most part, they are fine walking around on "coffee break", which can sometimes be more beneficial because sometimes they may come back feeling quite upset (usually after they die, for example), and when you zoom in on them, they might say something like "I need a break". Leaving them unassigned does in fact help this.

Alternatively, depending on their stats, you can send them to train, but obviously doing that once they reach max stats isn't going to be very productive.

You can assign them to the front entrance, but this isn't overly helpful. Especially since you plan on sending them out again soon anyway, and this also applies to assigning them to a production room - once they're gone, you'll have to replace them.

To keep things simple, I personally just leave them to wander the vault. That way I can find them easily, I don't have to worry about covering for them, and they generally are happy enough anyway (>55%).

If that still doesn't solve the problem however, here is a list of all the things that can affect your dwellers happiness.

Breeding

If you have a particularly grumpy pair of male and female Dwellers, pop them in the Living Quarters together and - as long as they're not related - the couple will cheer up plenty once there's another child on the way. Source

Of course you can put them in their correct room, and that will bring happiness up over time.