How can I use a "side sinatra app" when deploying?

I've got an ordinary rails app which now gets some widgets which are rendered inside a Sinatra app.

On my local machine I run rails s and ruby lib/sinatra.rb which then fetches stuff from the rails app. As a result localhost:3000 and localhost:4567 work fine together.

How can I do all this in a staging environment? The deployment itself is working and the rails app starts up. How can I access the Sinatra stuff now?

One idea would be mounting the Sinatra app to some path in the rails app? Any ideas, suggestions?