No question, that has been a hallmark of his regime. Signing our own won't take us out of the market though.
It's also important to remember that not many will actually get re-signed this summer. Belichick has never given out more than two large contracts in a single summer, and I don't imagine that will change. Two will most likely get extended, some others will play out their current deals. We will probably see the return of the franchise tag next summer as well.
We have a lot of options, but not all of them will be used. Some will be frustrated we don't keep more when we can, but BB will always keep flexibility as a core function of his management philosophy.