Cat peeing on rug under litter box... any advice?

I've had Brady for about 2 years now and he's never had any litter box issues.

I typically keep an old rug underneath the litter box so that if he steps out and gets litter out of the box, it isn't just on the floor.

Yesterday I was in the laundry room and caught him squatting behind the litter box on the rug, just peeing. 
I was upset and pulled up the rug to discover he had apparently been peeing on it for a while! What the heck!

So i washed the rug like 4 times and I couldn't smell the urine anymore so I put it back under the box, thinking maybe he was upset with me because it takes me a few days to clean his litter box sometimes. Telling myself that I needed to keep up on it more and that maybe he was peeing outside of it because it wasn't clean enough, so I cleaned it to perfection yesterday.

BUUUUUT today the rug smells like cat pee again. And the litterbox is still spotless clean and empty.

However, we recently got a new kitten (think 2 days ago) and they get along fine, but I have witnessed Brady peeing in the kitten's temporary upstairs litterbox. 

So he'll pee in the upstairs litterbox but pees on the rug downstairs by his regular litterbox.

I'm stumped. Is he just peeing on the rug out of habit, and now again because he can still smell his old urine?

Is it a UTI?

Any advice? I don't know how to resolve this.

Thanks for any help!

 maybe he was upset with me because it takes me a few days to clean his litter box sometimes.

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He's not upset with you. He's just avoiding a filthy bathroom. A few days is way too long of a time to go without cleaning the box. At the very minimum, once a day, ideally twice a day for one cat. 

So i washed the rug like 4 times and I couldn't smell the urine anymore  

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 BUUUUUT today the rug smells like cat pee again. And the litterbox is still spotless clean and empty.

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Did you use an enzyme cleaner? Only a good enzyme cleaner can completely remove the smell of urine. A cat's nose is fourteen times more sensitive than ours. They can smell things that we can't. Since he is probably still smelling his urine on the rug, he thinks that's his new bathroom.

So he'll pee in the upstairs litterbox but pees on the rug downstairs by his regular litterbox.  

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As noted above, dirty box and traces of previous urine.

Is it a UTI?  

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Not a bad idea to get that checked out, but from what you described, seems like an easy solution: daily cleaning of both boxes, and use an enzyme cleaner to scrub the rug. If there is another space in your home, add a third box. 

Definitely go get him checked out by a vet to rule out any possible medical problems, Such as a UTI or any inflammation. Then, if he has a clean bill of health, you may need to address any behavioral issues.

As for cleaning the rug, the possibility exists that by sending them through the washer and dryer the stains and odors could have been "set" in. But, there's always hope in my mind! You'll need  a enzyme cleaner. I use Nature's Miracle, I know some vets carry Urine Away or something to that effect, just something to break down the urine because it can be RELENTLESS. Even if we don't smell it, they will.

I hope this helps!