Are you quilting it? Or sending it out, to LA (long arm) quilter?
38.25" + 6" = 44.25" & 44.25" + 6" = 50.25" In a perfect world you would want your backing 44.25 x 50.25. When it is quilted, yes it draws it in, some. With only 41" and needing 44.25" IMHO I would add more fabric. Do you have any left overs from your piecing. How about splitting the the 41" and use some left over fabric to make it, you would have a stripe piece down the middle, large enough ... no trip to the store.
If you are sending it to a LA they need the 'extra' to help mount it. I usually add 4" when sending my quilts out. It depends upon what the LA wants (3 or 4). But the sad part is that when we square up our quilt, all that extra that we needed, end up in our scrap drawer.
I made some twin quilts for my 2 DGD's and had purchased the fabric out of state and DID NOT have enough. Like you close, BUT not enough.... so I sewed on a small strip. Made sure I talked to the LA, that I really did not plan on the extra strip to be in the finished quilt. She mounted it accordingly, when I got them back & squared them up... poof no one knew but me.
Good luck