The Bucks entertain Stalybridge Celtic on Saturday before travelling to face Stafford Rangers on Monday.
Celtic's squad features several members of last season's Buxton squad, and Hayward expects the likes of Stephen Brogan, Shane Killock, Sam Wedgbury and Grant Shenton would love to get one over their old club.
"Stalybridge have got some of my old players," Hayward said.
"Players I released. They'll want to get a result here, no doubt about it."
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Forty-eight hours later and the Bucks will make the trip to Stafford, where Hayward expects another battle.
"Stafford, they've had a good start to the season and my mate's gone in there, Dale Belford (as assistant to Matt Hill)," the Buxton boss added.
"He's a good non-league manager. They’ve got them playing and it's not going to be easy on their patch.
"They're going to be very direct and a good side. I think we've got a tough start to the season looking at our early fixtures."
But tough is what Hayward expects week-in, week-out in the Northern Premier League this season.
"There's no easy games in this league," he added.
"Also, as we've got a bit of a tag on our back, it sometimes makes it even harder for us.
"I think what we've done is, from when we came in, we've turned the club around from a bottom six team to a top two team. Everyone's tipping us and South Shields to win it.
"But there are other good sides in there like Scarborough, Warrington, Mickleover - a lot of other good sides."
Buxton came from behind to beat Mickleover 3-1 on Tuesday night.
Andrew Dales put the away side ahead in the third minute, only for Thomas Elliott, Christopher Dawson and Jamie Ward to respond with three goals in 13 second half minutes.
The Bucks drew 1-1 at FC United of Manchester of Saturday.
Diego de Girolamo netted his third goal in as many league games to secure a point in the 77th minute.
Aaron Morris had put the Red Rebels ahead early in the second half.