The Huddersfield striker agreed a three-month loan deal last week, but the Reds will wait a further seven days so he will be eligible for the rest of the campaign.
Spencer will therefore miss the game against Carlisle on Saturday but should be available for the trip to Sheffield United on January 26 if the 21-year-old overcomes his current knee injury.
Boss Richie Barker told the Crawley Observer: "We've put it back a week so that it ends on the final day of the season.
"We wouldn't half have egg on our faces if we find he has a lot of goals and we get to the last day of the season needing something and he's not available."
Barker also confirmed he has fielded "an enquiry" about striker John Akinde, currently out of favour at Broadfield after making just 10 appearances this term.