If you are bored and want to try something, these are areas I have found that "limit" the proper sizing expected.
Lube... alot of lubes don't work when you need to max out sizing. The RCBS liq in a small 2oz container is some of the best for this type of work... fairly thick and you squeeze onto your fingers to rub on the cases liberally. Unfortunately, some spray lubes just dont handle the pressure in the die.
I use alot of lanoline/iso alcohol lube but when I really get a tough project, I will use the RCBS.. some will use Imperial sizing lube.
Spring back in the press. Lots of presses have some slop at full travel.. ie you can push the ram "down" when at full stroke. If you only have the SH touching the die at full stroke without a case, it may be set too far away once you add the spring back under pressure. Consider turning down the die another 1/2 to full turn and the SH will stop on the bottom of the die before full down stroke.
If your press cams over, you may need to set the die so it touches 2/3's of the way down.
Lube well, run the case through, run through again and you should feels less resistance, run it through again till there is no resistance. Obviously lube the inside of the case neck very well too. With really oversized cases, you may need to work the case back down to min spec.. not saying this is good practise but it will work.
Lee FL sizers tend to be on the min side of SAAMI spec so if your sized case is still too big, odds are something is not letting the die do its job.... or you may want to check the headspace of the rifle in question.
From no fit to fit could be as little as a thickness of paper so it really doesn't take much to limit success.
New brass is the best solution BUT give this a try and if it resolves, you know a trick for future jobs.
Jerry