There are so many great options for back carries...these are just the ones I happen to have in my stash.
The woven wrap looks like a Moby wrap but isn't stretchy. It was pricy used, but I can back carry from 6 weeks. A wrap is the easiest to make, just cutting and hemming, but has the biggest learning curve for use. The hardest part about sewing a wrap is finding a good fabric! It can't be stretchy for back carries, and thin cotton kind of digs into my shoulders. The wrap I purchased is a Didymos that is made of woven fabric. It is very soft and strong. Some people use linen and make their own, but I haven't.
My Beco carrier is what we use most for Levi right now. We bought it used for around $35, and it has been an excellent investment. It has virtually replaced our mei tais since buckles are faster than straps. Also it has padded straps...when I made my mei tais I didn't realize how much more comfortable padded or extra wide (4inch or wider!) straps were!
My mei tais were what we used for back carries before we had the woven wrap or Beco. They are great, versatile carriers. I used this tutorial for my first one. It turned out pretty well, a bit wider than I wanted, and I angled the straps out a bit too much. Very usable but my second one was better. I have one a friend made based on my second one, I'd be happy to loan it to you for a few months if you'd like so you can try it out. It doesn't have padded shoulder straps, but I think that is the only change I'd make! You could easily add a few strips of fleece to the straps to add padding, if you wanted.
If you choose to buy instead, I've heard good things about this one and its pretty affordable comparatively.