You can always apply again for a different apartment. I would ask the question of why you weren't approved in tje first place.
If the board turned you down due to financial reasons (ie. liquid assets), yes, you can apply for a different apartment with a lower price point or lower maintenance fee.
If you had a dog and the board isn't accepting any more pets in the building, then that's a different case.
You can try asking the sellers broker or have your lawyer ask the board why you were declined. The coop board isn't required to respond however.
This may help you in deciding whether or not to apply for a different apartment.
Hope this helps.
I would advise you to move on to another building. Let me know if I can assist.