Want to extend fixed lease by 1 month

[Will Gates]

My tenant's fixed lease is up at the end of June.  They are not bad tenants per se, but, we inherited them, they've been in the house 4 years (we've had them for 8 months), they pay in cash now, they are below market rate, they keep adding pets, etc. - so we want them out. 

The problem is we have approximately 2 weeks of travel coming up for July, and we need time between their tenancy and the next to do some updates (paint, new flooring, new water heater), and I don't want the house to sit vacant while I'm out of town.  

Neither of us have broached the topic of renewing their lease term, so at this point I'm assuming they are planning to stay and would like some extra time, so I thought it might benefit us both to end the fixed lease on 6/30, and then start a new month-to-month lease with a 1 month term and 30 days notice, all on 7/1, with some language built in for their leniency on showing the place to prospective tenants.  

We live in a premier rental area and I'm not worried about a house vacancy in mid to late summer, as I should still be able to rent it quickly. 

I read elsewhere that others bump up the rent for the extended lease option, but as they can barely cover the rent as it is and I actually want them to stay another month, I'm thinking of leaving it as is. 

Thoughts, suggestions, and do you have any example language that you would put in an agreement like this?  Otherwise I'm thinking a standard 30-day notice "reminder" that their fixed lease is expiring, a new month-to-month lease for 1 month, and a 30-day notice for end of the month-to-month lease.  (This property is in TN where it's 30 days written notice required.)