Hi JuneBillie,
In regards to a fleece backing, yes, it will make it softer. But, omg, is it ever heavy. I am in the process of undoing a baby quilt that I started to quilt with fleece only on the back. (no batting) I dislike it intensely. I am really glad I stopped quilting it when I did. I feel like it cheats my beautiful quilt. And makes it look like a blanket. I put way to much work into it to just hate the look of it with the backing. That is just my opinion.
I have heard of putting flannel on the inside as batting. But, wash, wash, wash. Flannel can shrink a lot. Make sure you wash and dry it 3 or 4 times before using it as batting. As it can alter the finished look of your quilt. Sometimes, almost distorting the quilt. Again, just my opinion.
My favorite baby quilts, I made with a light weight batting and flannel on the back.
I make my baby quilts crib size (oblong) for the most part. I just google crib size quilts and work with those dimensions. Or, you could do something smaller to fit a stroller or do a playmat quilt size for the floor.