Pools' uphill season-long fight for survival ended last week when relegation was confirmed with Oldham's win over Yeovil, but supporters gave the players a rousing send-off after Saturday's televised final home game of the season ended in a 1-1 draw with Brentford.
"I have to say that the reception the fans gave the players after the final whistle gives us real hope and encouragement for the future," Green told the club's official website. "I can guarantee that everyone at the club will be working as hard as we possibly can to try and make sure we reward that support with a much more successful 2013/14."
Green added: "The way our supporters backed the team throughout the game was a credit to the club and I was so proud to be part of it on Saturday."
Manager John Hughes replaced the sacked Neale Cooper in November with the club already entrenched in a relegation struggle after going 14 league games without a win.
That winless run was extended to 20 matches before victory at Sheffield United at the end of December.
Although former Hibernian and Livingston boss Hughes steered them to a seven-match unbeaten run in February, Pools went eight games without further success last month and their fate was sealed.