Accies secured their return to the top division following a three-year exile thanks to a penalty shoot-out victory over Hibernian at Easter Road in May.
And now Neil is hopeful the team's attractive style of attacking play will tempt local supporters from the Lanarkshire area to sample the atmosphere of New Douglas Park.
Speaking to Sky Sports News, he said: "We've got to do that because we want as many fans as we can possibly get.
"There is always going to be a ceiling for us as our fanbase is limited, but the more we can attract then the better player we can try and get to come to the club, and the better the product the fans will see on the pitch."
Hamilton chairman Les Gray is relishing the prospect of the club punching above its weight again as they mix it with Scotland’s best teams.
Gray said: "We're the smallest club in the league but we've got a bit of attitude and we’ve got a chip on both shoulders about it and as far as we’re concerned, bring it on."
Meanwhile, Neil is upbeat on Hamilton's chances of making their mark next season, but refuses to set any definite targets.
"We are going to try and finish as high as we possibly can," he said. "Similar to last season, we sort of said top four, but let’s go and see what we can do."