Welcome to Wefugees and thank you for reaching out to us!
I'm not an expert on Finland's policies but had a look and found plenty of helpful websites around residence permits for work.
The first website should answer your question if you can extend your short-term visa: https://www.infofinland.fi/en/moving-to-finland/non-eu-citizens/a-short-stay-in-finland
You may have to contact the Ministry of Foreign Affairs/ Police (see website) for more information on your options.
Furthermore, here is information on the steps you need to take in order to apply for a work visa: https://www.infofinland.fi/en/moving-to-finland/non-eu-citizens/coming-to-work-in-finland
Please note that the website states you have to be out of the country in order to apply for the residence permit for work:
"If you come to work in Finland, you need a residence permit. Before a residence permit can be granted to you, you must find a job in Finland. When you have found a job, you can apply for a residence permit. You must apply for a residence permit before you come to Finland.
To work in Finland, you normally need either a residence permit for an employed person or some other residence permit entitling you to work. The type of permit depends on what kind of work you will be doing."
I hope this information helps. Feel free to reach out to us again if you have any more question, or reach out to a local immigration authority :-)
All the best,