How could I go to interviews during my resignation period without offending my supervisor or accepting my leave to go to interviews? Do I really have no choice but to tell her that I am resigning and would like to take a leave to go on interviews?
This is difficult, because you have already resigned, but that fact hasn't been announced to your supervisor. So your supervisor is rightfully expecting you to fulfill your contract and work, not take repeated times off. But in less than 3 weeks, your supervisor will know the truth one way or the other, and you will be gone.
Unfortunately, the interactions/notifications with the customer are the outsourcing company's responsibility, and not yours. If she only finds out about your leaving on your last day, she'll likely be offended anyway (hopefully in that case, her offense will be with the outsourcing company, and not with you personally).
If you don't want to offend her, ask permission from your employer (the outsourcing firm) to talk with your supervisor and let her know today that you will be leaving, and when.
If permission is granted (or if someone else in the outsourcing company informs your supervisor), then also ask your supervisor if you can take some time off between now and your last day to attend interviews. If your supervisor consents, then work with her to give her the hours she needs to transition away from you, and hopefully free up some hours for interviewing. That will avoid offending anyone.
If you don't have permission from your employer to announce your leaving to the supervisor, or if your supervisor doesn't give you permission to take time off from your contracted work, then you have no real choice other than to wait the 3 weeks and work out the remaining days. (If, as stated, you cannot interview after hours or on weekends.)
At that point 3 weeks from now you'll be unemployed and thus will have plenty of time for interviewing, and nobody can be offended.