The visa for the designated skilled labour has been defined to only one who has s status as ‘ mid-to-long-term residents’ in Japan.
Since 1st of April, any foreigner who has a residence status in Japan ( including refugee who has a Designated activities) are also able to apply for the examination of ”the Designated skilled labour”
First you need to pass the exams for knowledge of each majour genre and Japanese language.
When you pass the examination, then you have a chance to find a new job, then the your new employer should arrange your work permit.
However PLEASE REMEMBER the positive result of the exam doesn’t guarantee any of getting new job and work permit.