You can open the dashboard, search the script (you should choose a sort option which makes the script appear at the top) and then click at the "Last updated" column to trigger a new update check.

For development (and only development) you can also copy your script's header to a new script and then @require your script URL from your local web server.
Then open the TM settings page and set "Config Mode" to "Advanced, search for "Externals" > "Update Interval" and change it to "Always". Now TM will update your @require every time the script _was_ executed. Again, please use this only for development!

I hope (user-enabled) local file access will be implemented by the Firefox developers soon or at least one day. If that's done, then to @require a local file will make TM load it at every script execution without disadvantage.