Well, there's basically an adult (or, not aimed at children, at least) Reading Rainbow he does now as a podcast called Levar Burton Reads[1], so there's that.
Sadly those are from the 90s Reading Rainbow, not the original 80s series.
There are a bunch of seasons available free for Prime members on Amazon Prime Video. I've been watching it with my kids the last few weeks, and it's held up well!