Starting a lease mid-month?

[Matt Faix]

We had a tenant sign a month to month lease starting on Oct1 that goes until the end of the month and renews. We told her ahead of time though that there will be a contractor there the first week, and if she's able to wait until the 15th that would be great and we would prorate her rent. This was fine until her circumstances changed and she had to move in on the 3rd. It wasn't a huge hassle, but since her lease technically was starting on the 1st so we had to honor that. 

Is there a way to have a lease start on any day of the month (say the 15th), and still have it be month to month based on the 1st day of the month for all other months. This seems like it should be pretty straight forward but the way our lease is setup I'm not quite sure how it would work. Here's the wording we have. 

3. STARTING\ENDING DATES OF LEASE

a) This lease begins on 10/1/2015 . This lease ends on 10/31/2015 .

___X__This is a month to month lease

43. RENEWING LEASE

a) This lease will automatically renew for the same term as the original term if not ended or changed by either party.

Any help is appreciated. Thanks!