Amond, 28, scored 33 goals last season for Grimsby as they won promotion through the National League play-offs.
"There were a lot of clubs in for him but we're delighted that we've managed to tempt him to The Vic," said boss Craig Hignett.
Former Chelsea trainee Woods, 26, joined in 2014 from Harrogate Town and has signed an undisclosed deal.
He has made 42 league starts for the club over two seasons, having missed the latter stages of 2014-15 with injuries to his ankle and fibula.
"Michael is a terrific player and I am delighted we have managed to get him sorted," added Hignett.
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Hartlepool United have signed striker Padraig Amond and agreed a new deal with midfielder Michael Woods.