A long time ago, I wrote something similar ( https://github.com/htcat/htcat) to assist with Heroku's efforts to speed up moving the tar formatted application releases around. It's pretty old, it doesn't integrate tar archival itself, it probably can stand improvement. Or given its small size, even a rewrite.

My favorite hack in there (that also made it work with pre-signed S3 urls) was not using the HEAD method as is customary to determine object sizes, but instead doing a regular "GET" that, for small files, would execute on its own...but for larger would simply be abruptly closed by htcat once it reached the bytes that had since been fetched in parallel by a range-based request sent immediately afterwards. The goal was to have htcat not offer a penalty on small files so it could be used on blended workloads without thinking.

It also found a bug in S3's range implementation. We had a problem with some object or other, I wrote in about it, and was told that upon investigation a bug had been fixed. No more problem.

Some kind soul packaged it for Debian, so it's easy to get if you need it.