Long Term Luggage Storage - Paris

the hotel you stay in at the beginning if it is the same hotel as at the end, in their luggage room?

In case a hotel is not an option, there are several other choices (I have used the first two and can recommend them and the other two have been recommended to me by an American who lives in Paris):
Blue Marble: https://www.bluemarble.org/ParisLuggage.html
City Locker: http://www.citylocker.fr/site/files/
Holi Bag https://holibag.io/
Nannybag: https://www.nannybag.com/en/home

I think you'll find decent storage options, but in a pinch, you should be able to get inexpensive outer garments (tops and pants) at street markets if you're not too tall. I've bought cheap, lightweight cotton slacks (ethnic-looking prints) in a lot of places on the continent, including Spain and France, and of course tops are even more widely available. I've seen socks, too, and socks from a market couldn't possibly be worse than the misshapen things I bought at a Camaïeu store (avoid, at least for socks!). Undergarments I've never needed to replace in Europe.

One tip: Start out with new socks to reduce the likelihood that yours will wear out mid-hike. I guess that's a mistake a serious hiker would not make, but I made it.