The Ayrshire side lost their Scottish Premiership opener 2-1 at Rugby Park.

But Alessio, who succeeded Steve Clarke this summer after last season's third-place finish, said: "We still have to improve, but I'm happy with my team.

"I've read the players are unhappy, but that's not true and they showed it with a good performance."

The Italian started his tenure with a shock Europa League exit against Welsh part-timers Connah's Quay Nomads.

He has since had to respond to reports of behind-the-scenes crisis talks between the squad and Kilmarnock owner Billy Bowie.

"I'm sure of my team and my players," Alessio said. "These players have a good attitude and mentality, which they showed in every game last season."

Alessio said last week that he did not have enough players in key positions to implement his preferred style of play.

Kilmarnock have only made two summer signings after eight who were regulars in Clarke's squad left in the summer and the manager stressed that it was important that players, fans and everyone should "stay compact together" at this time.

"We have a lot of conversations," he told BBC Scotland when asked if more signings were on the cards. "Of course, we need other players and we need to complete the squad and I hope that, in the next days, we can conclude some deals."

Alessio said Kilmarnock would have to work harder at defending set-pieces after Scott Arfield and Connor Goldson scored from corners at Rugby Park.

However, he was pleased at the way his side came back to level through Stephen O'Donnell's equaliser before losing to a late goal.

"We needed to press more and, in the second half, we showed a good reaction and played well," he added.