In normal play, during combat and then into the next combat after, it is not a problem. Some people here are specifically talking about the exploit of the PC waking up in the morning, casting Hex on a chicken, then having their own little short rest to regain the spell slot, before killing the chicken and holding concentration on Hex until a real enemy comes along to transfer it to.
Now I see why so many DM are against this shenanigan. It would not happen at my table.
PC Warlock: I cast Hex on my chicken.
Me: Ok. Done. What's next?
PC Warlock: I take a short rest to recover my spell slot.
Other PCs: NO!!!!!!
PC Warlock: Why? It is perfectly legal to do so...
Other PCs: No. He'll roll on the random encounter table. You don't want that. Ever. No way. We go right now as it is supposed to be.
PC Warlock: What is so bad about a random roll?
Other PC: You don't want to know. You don't want to risk it unless absolutely necessary.
Me: I simply smile and say: You're sure now you don't want to take a small ipsy tiny short little rest?
Other PCs: No! We press on. If he wants to rests, may he die alone.
In my game, any rest below the four to six encounters per day means a random encounter roll. The more encounter you had, the less dangerous the encounters are. They all know it from the start. It is not an auto encounter but starting with a deadly encounter at the start of the day is not a good idea at my table. This is why the problem never occurs.
As I said earlier, maintaining concentration on the spell does not requires it to have a target. Only the initial casting needs one.