The issue is that in order to give you a specific seat, they need it to be free. If the better seats are already given away, normally the people who have received a seat have this printed on their (e-) tickets. Moving someone away from their seat because you want it is very tricky.
So the best strategy would be to get a better seat in the first place instead of upgrading. Early booking, higher loyalty program rank and online booking are ways to do so. Some loyalty programs let you specify preferred lateral locations, too. Choosing an aisle seat is often better than being wedged into a corner. I prefer a window seat so I can lean against the wall and sleep.
The next step, if the above does not work, would be to check in as early as possible. Often you can pick your seat when checking in online and the sooner you do that, the better it is.
The next level would be to ask at the check-in and then again at the gate to make sure that in case there is a no-show, they would give you that seat. The issue with that is that you will be essentially the last guy getting on the airplane and then also likely have less chances to put your luggage into the overhead locker since it might be full already.
It surely differs by airline, but I would assume that in most cases the person who checks you in is not able to move someone else off their reserved/assigned seat without the approval of a supervisor. So unless you can give a good reason to get for getting an extra legroom seat (like a broken leg, a baby with you etc), it might be difficult to convince someone to change the seating order for you.
If there is no way to reserve your ticket at all until you are at the gate, the goal is to be the fastest. Check in early, stand at the gate. Having kids that can run ahead and grab a seat also might be a good strategy.