It is in Jackson's best interest to get a new deal in the first few days of the tampering period whether it is with the Patriots, with another team if the Pats decline to franchise him, or if the Pats decide to tag and trade him.
If the Pats tag him and intend to try to trade him, it may be in his best interest to sign the tender immediately because the Patriots are then on the hook for the money even if they cannot trade him (if he doesn't sign the tender, they can cut him without any cap implications or owe any of the money. That gives Jackson leverage in where he goes especially if two or more teams are vying for him and one will give him a better deal but the other is giving more in trade compensation or if only one team is interested in and he doesn't want to go.