The new lease cannot begin until June 6, 2012.
If you wish to have payments on the first of the month, then prorate June 6 to June 30 and then require full month payments on the first of each month after that.
Remember that leases for more than 365 days must be in writing and signatures witnessed by two witnesses. You can keep it simple by always keeping the lease to 365 days or less, so make the term from June 6, 2012 to May 31, 2013.
Of course, you must give proper notice to the tenants regarding the rent increase. I would suggest you send them a notice now certified mail to let them know of the rental increase.
In the renewal of a lease, the improtant thing is when the renewal term ends. If it is a renewal, it begins at the expiration of the original lease.
Disclaimer: This answer is provided for informational purposes only, does not constitute legal advice, and does not create an attorney-client relationship. Actual legal advice can only be provided after completing a comprehensive consultation in which all of the relevant facts are discussed and reviewed.
Review the terms of your previous lease agreement. You may be able to stipulate to a change in the terms, in writing, of your previous lease agreement so that it terminates May 31, 2012, and your new lease agreement runs from June 1, 2012, through May 31, 2013.
You should consider retaining the services of a landlord-tenant attorney to assist you in this and other matters you are certainly to run across as a landlord. There will be more complex issues that you will have to deal with as a landlord, you might as well take this opportunity to interview a couple of attorneys to see which one you find qualified, reasonable, and accessible. A good landlord-tenant attorney will make your life easier and let you rest at night. You have invested in real property, and you certainly want a qualified attorney in your corner to make sure that your interests are protected and that your lease agreements are drafted properly.
No communication resulting from this posting will create an attorney-client relationship. In order to create an attorney-client relationship you will need to meet with the attorney and execute a written retainer agreement. Please do not send any confidential information to us until such time as you have executed a retainer agreement and an attorney-client relationship has been established. Furthermore, this posting and any information contained herein are intended for informational purposes only and should not be construed as legal advice. Seek competent counsel for advice on any legal matter.