The Toffees looked favourites to land fourth spot at one stage before defeats to Crystal Palace, Southampton and Manchester City saw them falter at the last as Arsenal nipped in.
Everton have been to Thailand and Austria in pre-season and are building up to their first match of the season away at newly-promoted Leicester a week on Saturday.
Forward Naismith said: "We've done a lot of the fitness work and now more tactical stuff that the manager wants to work on to add more strings to our bow for the season.
"We've got a great squad full of fantastic players and we're gelling every week we're together under the manager.
"We've had a full year of working under him and knowing what is expected so we'll hopefully push on and better what we achieved last season.
"We definitely want to do that but we'll not look too far ahead. First of all you want to start the season well with some good performances and results.
"We were so close to Champions League last season and this season it would be really nice to have the same fight again and maybe make that fourth spot, if not higher."
Manager Roberto Martinez's most significant additions during the transfer window have been two of last season's stand-out players.
Gareth Barry arrived on a free transfer from Manchester City while last week Everton smashed their transfer record by paying Chelsea £28million for striker Romelu Lukaku.
It was a real coup for the Toffees to land one of the hottest young striking talents in the Premier League and a significant outlay for a club regarded as frugal.
Naismith said: "It shows we definitely mean business and there is a project under way. The manager and the board and the chairman are all pushing in the same direction, they're willing to back the manager in the transfer market.
"Gaz Barry and Rom showed what they can do last season and to have them tied down to permanent deals is something that's great for the club and for the squad as a whole."
Both players spent last season on loan at Goodison Park, with 21-year-old Lukaku netting 15 goals in 31 Premier League games.
The pair will look to pick up where they left off, and Naismith added: "It definitely makes it much easier.
"They know the way we play, the systems we play, and if we have to change during a game, they've experienced it all last season. That will definitely go a long way to help us.
"We've signed Mo Besic as well, who looks another fantastic player and I'm sure he'll have a very successful career."
Another statement of intent came with Ross Barkley signing a new four-year deal.
After a breakthrough season that saw him earn a trip to the World Cup with England, all the biggest clubs in the league were rumoured to be chasing the midfielder's signature, but Everton have held onto him at least for now.
"He's a fantastic player now with a lot of potential but the biggest thing he has going for him is his attitude," said Naismith. "He wants to learn, he wants to work hard and he wants to improve as a player.
"John Stones is another one who had a fantastic season last season for somebody so young.
"He was unfortunate not to go to the World Cup but making his international debut was an incredible achievement for him and I'm sure he'll want to push on this season as well."