If you don't have the option to add a parameter to the procedure to specify the date, your only option so far is to set the date in the job date table and run the procedure again.
But because of course I do not know the context in which the logic is executed and which additional steps are done based on the data (in the job date table and produced by the procedure) I cannot recommend it to you to go that way.
If you consider it to do you have to carefully check all dependencies.
The best solution would be to clarify the issue with your dev team (or the responsible unit) to create an option to fulfill your requirement.
Regards,
Florian