But he is hopeful of appointing a replacement - and filling two other key roles - within days at the Wessex League Premier Division club.
US were already hunting for a new head coach, to replace Steve Weston, who took a back seat from the role earlier this year. They have also been searching for a fitness coach.
Brady and other club officials are planning to meet on Tuesday to discuss which names could fill those positions.
Regarding Grice’s decision to quit, Brady - the US secretary - said: ‘It was a massive blow, particularly as Tom had been part of the discussions we’d been having.
‘Tom’s got a young family, his daughter is more active than she was a year ago, and he’s working more hours as well. Obviously family comes first, so something had to give.’
Under Grice, US finished fifth bottom of the Wessex Premier in the club’s first ever taste of step 5 football. Given the mass exodus to Moneyfields this time last year, Brady was happy with how the season panned out.
‘We probably exceeded expectations,’ he said. ‘We wanted to consolidate. We knew we were unlikely to be top 10, but we wanted to finish between 10th and 15th and that’s what we did.
Jeffes, who combined his playing role with the position as No 2 to Grice, is staying on. ‘He’s an integral part of the club,’ said Brady.