A FlyerTalk Posting Legend
Join Date: Sep 2009
Location: Minneapolis: DL DM charter 2.3MM
Programs: A3*Gold, SPG Plat, HyattDiamond, MarriottPP, LHW exAccess, ICI, Raffles Amb, NW PE MM, TWA Gold MM
Posts: 100,404
IME the grace period tends to apply only to rentals in the USA and airport locations use the time that you leave the gate as the start of your rental contract. Also, if you're too late (two hours?), the reservation might be cancelled and they could generate a new contract at the current rates if you stand in line and cars are available.
OP seems to be renting from National, but we don't know the country and whether it's an airport or city location. Also, OP has status with National which might make a difference.