Carver joined Steve Clarke's coaching team early last season and saw Scotland grow in confidence ahead of their play-off success against Serbia in November.
Now he believes there is the talent for Scotland to break new barriers after qualifying for their first major tournament in 23 years, by reaching a knockout stage for the first time.
Speaking two days before their Hampden opener against Czech Republic, the former Newcastle coach said: "What we don't want to do is just arrive at the tournament and say, right, we are happy to be here.
"We want to take the next step. Because if we have a good tournament, that will put us in good stead for the next part of the World Cup.
"This group of players want to compete and progress. I know the manager has talked about, every time we play an international match, they gain experience, they will get better and stronger, and that's what I am seeing.
"Since I arrived in September, I think we've had five camps and the quality in the group has got better and better and the standards have got higher and higher.
"I'm expecting us to get out of the group. The guys are confident enough to do that and if we can achieve something that's never been achieved by a Scotland men's national team, that's what we're looking for."
Carver believes the pandemic-enforced delay to the Euro 2020 finals has helped Scotland and pointed to the emergence of three players who won their first caps in the warm-up games after benefiting from the expansion of squad numbers to 26 players.
"Just look at the three young guys who've come in – Billy Gilmour, Nathan Patterson and David Turnbull," he said. "They've added quality to the squad.
"When they were named everybody thought we were just bringing three young lads in for experience, but let me tell you, these guys have put pressure on some of the senior guys, so that period of time has certainly helped us."
Carver has been working on further building the confidence within the squad but feels other key elements needed for success are present in abundance.
"One of the main things for me is camaraderie and togetherness," he said.
"We know that these guys have got ability. They have got to believe in their ability and I keep telling them how good they are, because I have seen and I have worked at the top level of English Premier League football, with top players.
"And these guys have got to start believing that they are good enough to go to the next level. I am seeing it every time they go on the training ground.
"But the biggest thing for me is the togetherness. Everybody wants to be part of it, everybody wants to be here. Nobody wants to miss out.
"The guys who unfortunately missed out will be devastated but they are still supportive. I know they have been in touch with some of the guys, because they want to be part of it, and they have played a part, some of the guys who are not here. And we must not forget them."