I think a bit of distortion, and some short delay and reverb to add that extra tapey fuzz would add that missing bit!
That’s what I pretty much did with mine. I was out of steam toward the end, so I didn’t really put much effort into the reverb, but in listening over and over to BoC’s version I thought that some of the lush graininess was coming from that type of processing in their signal chain. And of course tape sound…!