Trying to move to Sweden, need help figuring out visa

I am a young cyber security professional living in Estonia. I am originally from the United States I originally moved to this country as a student. After graduation I obtained a five year residency permit. However my current goal is to move to Sweden to live and work there. I have been applying for jobs there for the past two years but have not made much progress. I am worried that I have been taking the incorrect approach but I am not sure.
I have several friends who are not EU citizens who have been able to relocate to Finland and Sweden and are now working as cyber security professionals in those two countries. However I have heard several hiring managers tell me that their company policy prohibits them from hiring people who do not already have a residency permit. This does not make much sense to me, both since I have friends who did not have this issue, and also since it presents a catch-22. I cannot get a residency permit without having a job offer and contract, but these companies say they will not offer unless I have a residency permit.
I have been told about one potential workaround. My visa allows me to stay in another Schengen countries for up to 90 days. I may work in that country during that period as a temporary contractor, but any longer requires a residency permit. It would supposedly be simpler to sign a contract with a company for this temporary period and then have it extended, after which I apply for the permit. However this sounds kind of tricky and again, none of my friends had to jump through this hoop.
I wanted to ask if anyone here had any advice, or how may I change my approach?