This depends on exactly how you are streaming the cover song in question. Here's each possible scenario and how you should proceed.
CD Baby digital streaming partners
You DON’T need streaming licenses to distribute your cover song to our streaming partners (i.e. Spotify, Apple Music etc). Those companies are responsible for paying out all royalties to the publishers themselves. Our artists are NOT on the hook for royalty payments on those kinds of streams. When purchasing a license, there is no need to purchase one for streams. But you DO need to purchase at least digital download licensing to move forward with distribution, regardless of your level of distribution.
Personal website (SoundCloud, Facebook etc)
If you wish to offer streaming of your cover song on your personal website; or want to post a full song stream of a cover on Soundcloud, Facebook, etc; then you would need to secure a streaming license for your cover song.
Streaming covers on YouTube
On YouTube the presence of a video image makes it a synchronization issue instead of a reproduction, so you need a sync license. This needs to be requested directly from the publisher. Without it, you will likely get either a claim on your video via the YouTube CMS, a takedown notice, or in the worst case, sued.
The option is open to any artist to attempt to license directly from the publisher for all of the potential rights they may need, including sync for YouTube, etc. This is not practical in most cases, but there's no reason not to try, particularly for a sync license. It never hurts to ask but we cannot assist in this process.
Where to get a license
If you would like help reaching out to the publisher of the song to obtain a synchronization license for your cover song, Easy Song Licensing can help.