The striker had surgery yesterday to insert metal plates into his jaw, which was broken during Thursday's pre-season friendly against an Arsenal XI.
Clare – who has twice won the golden boot and Conference championship – became Hill's first summer signing as Diamonds aim for the Blue Square Premier title.
How bad a loss do you think Daryl Clare will be. Click here to have your say or post your comment at the bottom of this story by registering at the blue button.Hill said: "It was quite a bad break.
"This is a massive blow for the club to lose a 25-goal per season striker a week before the start of the league campaign.
"Daryl is a proven scorer year in, year out at this level.
"But these things happen in football. We have to move on now.
"It's disappointing but we will have to get on with it and be positive. We hope he will be back as quickly as he can.
"All of our forwards have done well in pre-season – especially when you take into consideration the quality of our opposition in those games."
Clare's replacement alongside Lee Phillips in the Diamonds attack is likely to be Dean McDonald.
He has scored five times – including a hat-trick on his debut – since joining from Inverness Caledonian Thistle.
Diamonds kick off next Saturday with a trip to Eastbourne Borough.
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