Agree with others that I’ve never had any issue using the SkyClub on arrival. As long as it is in conjunction with same day ticketed travel on Delta and you have a boarding pass, you should be good. Also note that if you’re in an airport with multiple clubs, you can generally use the receipt to go from club to club within the same airport if you’re getting bored and wish to sight see (or get a real meal). At least it has worked this way in MSP and DTW for me.