Like has been previous answers stated each application to rent will depend upon the actual owner(s) or management company will be making a final determination.
I see two positives from your statement;
1.) you dont owe any previous landlord(S)
2.) you been late but paid.
I believe everyone will understand your situation, some more then others. offer an additional security deposit or pay an additional month of rent in advance. A less expensive way be to have agreement to use some of your contractual skill to the table.