The Economist often uses "Better still". It is definitely not incorrect. Here are some examples:
"Better still, Babylon is capable of translating completely novel sentences." "Since then some figures have got better still." "Better still, Mr Blair was a true believer, exuding conviction."
"Better still" and "Better yet" are both used idiomatically to mean "Even better than what was just discussed." There will be a lot of chocolate at my birthday party, and better yet, a sundae bar.
"Still better" would be used as follows: I'm not very pleased with this old dress, but it's still better than the ones I'm seeing in the stores now. Or: I know that Dean Martin songs are "oldie