Merle, I did not know for sure what the value of your barrel is so I did a little internet searching. The big auction site had two past sales, one sold for $152 and one for 220 with forearm and target sights and listed as very nice condition. Ed's contenders shows one for $145.
So there does appear to be some interest in that caliber barrel.
I personally don't think "New in box" and "hardly shot like new" would make a big price difference on a Contender barrel except for a collector who wanted an unfired one to add to his stash.
Value sometimes is determined by who wants it and how much they are willing to spend, so you might ask a really high price and the right customer might jump on it. You just never know.
Good luck and enjoy your barrel.