If your old domain has been widely used and promoted, then presumably it has accrued back links, been mentioned elsewhere online, and will be recalled by people who have encountered it before. So there may be some residual value in those associations and even some inbound traffic.
Selling the domain would mean losing that. Since the domain itself may not attract much bidder interest and the sale price (if there is a sale) might be disappointingly low, you should consider whether it isn't better, after all, to maintain the current domain and have it forward automatically to the rebranded version of your website.
I agree with Joseph. You probably have "Google juice" you don't want to lose. You should look at your analytics and see what the top URLs are for your existing referrals and inbound links and then set 301s for those on your new domain. I just had a client do this with thousands of links. The old domain then stays in the portfolio for awhile. After all the 301s have taken effect it could be worth selling at that point. If you also have collateral assets such as branding, design guides, logos, social handles, etc. you can package those up. I've seen packages do better than just domains because you can pitch the entire marketing wrap to the buyer. I sold one of mine for several thousand dollars for a very generic domain. However, I will warn you that you need to use reputable services because I had a client who did this on his own once and it turned out that they were scammers. You have to be careful because it's still your old brand and people will track you down thinking you scammed them. He dealt with that for months and ultimately had to remove his own company from his LinkedIn. It was a mess.