Saturday saw the men’s team dump National League South side Chippenham Town out of the FA Cup in the third qualifying round after a 1-0 away win.
Kenny Pogue netted what proved to be the winner shortly after the break, rifling home into the top corner.
The only sour note was a straight red for Ben Pope with 10 minutes remaining.
Hastings have been drawn away - for the fifth consecutive time in this season’s competition - to National League side Maidenhead United in the next round on Saturday, October 16.
Wood said: “I’m very proud but not surprised. I’m very bullish and confident in what we’ve got at Hastings.
“Just the manner and the way we deal with things gives us the edge to be able to compete higher.
“I’m massively proud because we showed that this club should be operating at a higher level.
“Every time we get the opportunity to play a team that’s higher, we never embarrass ourselves.
“We should have beaten Weymouth [in the FA Trophy in 2019] and we were very unlucky not to win at Ebbsfleet [in the FA Cup] last year.”
Just a day later, the women’s team booked their spot in the third qualifying round of the Women’s FA Cup with a 2-1 victory at Islington Borough.
Molly Hill gave the U’s a 12th minute lead before the hosts levelled a minute later.
Georgia Tibble was sent off for Hastings just before the break, but Hill netted on 50 minutes to book a trip to Bromley on Sunday, October 24.
Wood added: “In seven games, we’ve had six wins and one draw. But you’ve still got to do the job, and, once again, the girls did.
“I’m very proud with the fact that they came back after having 10 players on the pitch after an early red card.
“But they never looked like they had 10 on the pitch. They always looked like the better side and as the game grew they got better and worked hard for each other.”
The men host Isthmian South East rivals Ashford United in the FA Trophy second qualifying round this Saturday, while the women welcome Maidstone United in the London & South East Regional Women's Football League a day later.