Town are in the last 16 of the competition following a comprehensive 4-1 win over Sky Bet League One side MK Dons.
The Glovers face Reading Under-23s at Huish Park on Tuesday (7pm kick-off) and a win would see them just three steps away from Wembley.
“I’ve tried to really support it as a manager,” Way said in part three of his Half-Way Report on YTFC’s official YouTube channel.
“We haven’t had great crowds but I feel if we think a little bit different, knowing the reward at the end of it, why would you turn that down?
“I feel if I support it then the players will start to believe a bit more.”
Yeovil have already beaten Reading in the Checkatrade Trophy group stage this season, thanks to goals from Tom Eaves and Francois Zoko.
Way believes the young Royals put a strong side out on the day and expects them to look for revenge when they visit Somerset.
He said: “The majority of the teams have played the best teams that they could against us so I’m expecting Reading to put their strongest side out.
“The players who played in the previous game at Reading were a strong team so we can’t underestimate them.
“Just like the Crawley game they’ll be looking for revenge so we’ll make sure we do our homework on them.
“We have to make sure we prepare right, which we do for every single game, and if we win it’ll be another step closer to Wembley.
“The supporters have been great and I’d love to get as many through the gate as we can to try and edge us that step closer.”
Junior tickets are available at just £1 for the match with adult tickets priced at £10.
Fans can purchase tickets in person at the ticket office, online or by phone on 01935 837888.