What the law is and how the JPs follow the law are sometimes two different things.
In my lay person's opinion you have changed the terms of the contract/lease. You've now extended a month. So I would think yes. Some JPs don't like Pay or Quit. They just want to see a Quit letter. So in my opinion you would want to send another vacate letter (Quit). The way I read the law is you must give them 3 days, you can send them the notice or post it inside the back of the front door. I normally send regular (the law), certified (interpreted by JP) and post on the door and take a picture. If they are at home I personally hand it to them. Then you have all three methods covered. You have to give them reasonable time to have received the letter and have their 3 days before filing. So if you hand it to them personally, you could file after the 3rd day.
I've never seen one be able to pay 2 months, if they missed 1month, so get ready.
I would also talk to them. Be empathetic. If they can't pay, they need to move. Tell them you don't want to evict and eviction is very bad on their record. It will haunt them. So the best thing to do is move NOW and leave the place clean and in good condition so they can get their deposit back. However if you file, follow it through, even if they move in the process. That way you can get a judgement. Sometimes they will pay when they want to buy a car or a house in the future or get a job that checks their credit.
Here is the best advice you can receive when dealing with Real Estate Contracts : Contact a Texas Certified Real Estate Attorney.