The Cardinals also got a fourth-round pick (for Tony Jefferson) and a seventh-rounder, while the Texans received two sixth-rounders.
The NFL awards compensatory picks based on free-agent departures the previous spring.
The formula considered the Patriots losing LeGarrette Blount, Chris Long, Barkevious Mingo, Logan Ryan and Jabaal Sheard, and adding Stephon Gilmore, Rex Burkhead and Lawrence Guy. The Buccaneers also have their own picks in the first six rounds plus Pittsburgh's sixth-round selection - courtesy of last summer's J.J. Wilcox trade - and the announcement of the compensatory selections means they also know exactly where those picks fall.
The Minnesota Vikings were the only other team to receive multiple picks, with both coming in Round 6.
According to the league's release, compensatory free agents are determined by a formula based on salary, playing time and postseason honors.
A year ago the Browns were awarded 4 picks.
The NFL did not count the team's signing of Martellus Bennett in the calculation of the draft pick distribution. Cornerback Brendan Langley (No. 101) and quarterback Chad Kelly (No. 253) were chosen with two of the team's four compensatory picks in 2017.
The Giants lost Hankins, kicker Robbie Gould, offensive lineman Marshall Newhouse and defensive back Cody Sensabaugh last offseason. The Cincinnati Bengals, Dallas Cowboys, Green Bay Packers and Oakland Raiders all were awarded four, which was the most in the league.