I have a map that weighs almost 1 TB on the server. Unfortunately, my server is too weak to generate new fragments, so I would like to generate them on my computer.
I cannot download the entire map because I do not have so much space on my disk, so I would like to generate only new parts of this map (.mca files) and then copy them to the server.

How to make the plugin start generating selected fragments right away without verify the rest of chunks?

You can increase the radius of the existing task that finished (and set to uncancelled) or create a new task and set the chunk count to manually skip the inner portion. If you do this while the server is live you may need to use chunky reload before continuing the task. If you need any further help with that please join the Discord server since it's easier to chat about there than through a Github issue.