Gap Between Move Out Date and Lease Start?

Your lease should have started on Sept 30th so you avoid the gap. If the place is not ready then you should have negotiated for Oct 31th move in. Unless the touch up is major, maybe you can ask your new landlord if you can move in on Sept 30 and have the people do the work while you are there. Otherwise, crash at your friends or pay up for a hotel. If you can't move in due to repairs, make sure your landlord is pro-rating your rent to reflect.

Ask your landlord if they can do an extension for a week. Do this without signing a new lease though.
We had the same problem and asked our landlord for an extra two weeks And it worked out. Good luck

If you have a lease for 10/1 and there's no stipulation that it might only start on 10/5, the apartment needs to be ready on 10/1. If it's not going to be ready then as others have mentioned you need a hotel or some other bedbug infested facility for those days (and if it's the new landlord's fault you may be able to recoup that cost). Another cost, if you're using movers, is that they're going to charge you extra for getting your stuff in and out of storage.