You could use a pid file for that. This is, as far as I know, the mechanism commonly used by server software.

At start up, Your script checks if there is already a file with a given name, usually yourscript.pid in a directory, usually /var/run. If there is one it may read the contents which would be the process id of the already running instance and print an error and exit.

If there is none, Your script would create yourscript.pid and write its pid to it. Before the script exits, it deletes the file.