it is correct that we on this page https://www.a-kasser.dk/benefits.html tells that is should be possible to get Unemployment benefits ("dagpenge") if you are encompassed by the "Job change rule".
The "Job change rule" is described in english here: https://www.nyidanmark.dk/en-GB/Words%20and%20Concepts%20Front%20Page/SIRI/Job%20change%20rule.
The job change rule makes it possible for you to start your new job before you have received a new residence and work permit, i.e. you can begin your new job once you have submitted the application for a residence and work permit based on the new job.
If SIRI has given you a job seeking permit because you have lost your job under the Fast-track scheme, the Pay Limit scheme or the Positive list, through no fault of your own, then you will also be encompassed by the "Job change rule". You may therefore begin to work once you have submitted the application for a residence and work permit.
Eligible to receive dagpenge during the job seeking permit?
If you can start in a new job with one days notice (which you can according to the above mentioned, if you are encompassed by the "Job change rule"), then you are considered available for the danish labour market. And therefore you meet one of the requirements for being eligible for dagpenge.
For being eligible for dagpenge, you must off course meet all the requirements, which you can see here: https://www.a-kasser.dk/benefits.html.
However, this question / requirement is subject to uncertainty, as the point of start is that a jobseeking permit does NOT allow you to work. And you cannot see any public site/sourses where it states that you should be eligible for dagpenge during a job-seeking period.
However, we have write the above mentioned paragraph on our site, because it is our opinion that you meet all requirements for being eligible for dagpenge if you are encompassed by the "Job change rule".
Today we have send a question to star.dk to get this confirmed, however answers from Star.dk can take up to 6 months, so we can not help you further in you case, but will update our website as soon as we know more.
Until then, I will recommend that you try to apply for dagpenge in you a-kasse, and if they refuse to accept your application, then you could try to use some of the things/arguments we have pointed out in this answer.
I would look foreward to hear the answer from your A-kasse.
Anders Weber, www.a-kasser.dk