I don't think there's any extra effort required. Players normally sign with the team that offers the most money.
I disagree.
Imagine being a twenty-something millionaire: I hope you can see the appeal of a large, warm weather city v. a small, cold weather town.
Now consider the fact that many warm-weather football cities in the south (e.g., Atlanta, Charlotte, Miami, Jacksonville, Houston, Dallas, etc.) boast much friendlier income tax rates. If I were an agent, I would make sure my clients understood that $20 million in Houston means a LOT more take-home money than $20 million in Green Bay. (In the NBA, the Toronto Raptors have to deal with even higher income tax rates in Canada and it's a negative when the franchise tries to attract FAs.)
As an added bonus, there may be no need for a second home in a warm weather city during the off-season. Since, if I play for Green Bay, it's pretty much a given that I'm not living there during the off-season if I have the income to enjoy a more pleasant climate elsewhere.
This isn't to say that the Packers can't find FAs that want to suit up for thirteen-time world champions. I'm just being a realist about the factors that might discourage FAs from signing with a team like Green Bay. Personally, I love Wisconsin.