Is there a better way to find sponsorship opportunities? (USA to Scotland)

Hi,

I'm a professional software engineer from the USA hoping to immigrate to Scotland. I've been going down the 2,000 page document containing the list of companies that do sponsorships and sending my CV to the companies doing the kind of work I'm qualified for (applying to postings where applicable), but I'm wondering if there's a better way to go about connecting with companies that might offer sponsorships.

Thanks.