Backend App is Done, Published using single instance of VM, A Guide to Scale?

So here's the thing. There is a backend application that contained REST API service. It was already deployed in a VM instance.

Horizontal Scaling
In order to enable the backend app to serve more connections, is load balancing (instantiate new instances and config NGiNX) the only thing possible?

Vertical Scaling
Alternative way to enable the backend app to serve more connections, is upgrading the instance spec the only thing possible?

Other Scaling
Is there something other than horizontal or vertical scaling in order to enable this backend app to serve more connections?

Something like refactor code to be more efficient by add or change something in the code?

P.S.
I appreciate any kind of thoughts about this topic.