How about trying for a lower level card like a TD Bank,Citizens or Key Bank credit card?
The cards you are looking at are for people who already have 1. let's say for example somebody has a Wells Fargo Visa but got denied a Chase Freedom. They might want to try for a Bank of America instead.
You who are a 24 year old with a part time job, got denied for a lower level Chase, the Slate card. The only thing I would suggest is you could ask your dad if he can go to the bank with you, have him talk to to the person that spoke with you, see if he can co-sign for you.
Or try for a Orchard credit card
these would be my suggestions to you.