The capital side won only three of their last 10 league games to slip to ninth place last season, having begun the new year in the top three.
"Last season wasn't acceptable," said fit-again lock Fraser McKenzie.
"We are really looking to achieve top four. I think that is the minimum criteria this year.  We really feel we have a good enough squad to compete."
McKenzie, 28, is enthused by the influx of new blood this summer, with three newcomers - fly-half Duncan Weir, full-back Glenn Bryce and wing Rory Scholes - starting Friday's pre-season match against Newcastle at Murrayfield, before their Pro 12 campaign begins at Cardiff Blues on 3 September.
"We have made some really good signings who have bedded in well and quickly," McKenzie noted.
"Competition for places is at a maximum at the moment and it is all looking really positive for this season."
Edinburgh managing director Jonny Petrie revealed on Thursday that one more new player is set to arrive "at the start of next week".
But along with McKenzie, who missed the second half of last season after an operation to remove a disc in his neck, the return of fellow lock Grant Gilchrist is akin to another new signing.
The 26-year-old suffered a second broken arm in February in his first game for the club since October 2014, when his first arm fracture occurred.
"He was in a bad position but how he has come back from his injuries is really impressive," said McKenzie, who spent the latter part of his own rehabilitation programme working with Gilchrist.
"Grant has been made joint-captain (with hooker Stuart McInally) and has a unique set of leadership skills.
"He is very good at encouraging his team-mates but he is also an incredibly big man, a tall guy with good line-out and aerial ability.
"He is an 80-minute rugby player, even though he is big, with good fitness and skill levels - he has a hell of a lot of ability to push further forward. He was in the mix to be Scotland captain before his injuries and has the ability to get back to that sort of level."
Gilchrist, a target for French club Toulon before agreeing a new one-year deal, is one of several internationals - alongside prop WP Nel and flanker John Hardie - who are likely to be subject of interest from England and France with their contracts up at the end of the season.
While acknowledging Edinburgh "don't necessarily have the financial firepower to operate at the top end of the market", Petrie remains optimistic of retaining the club's leading lights.
"WP [Nel] is one of the most dominant tight-heads in world rugby and likely to be a good candidate for the Lions as well," Petrie noted.
"I don't doubt there is going to be interest from other clubs across the world.
"But WP has been with Edinburgh for a few years now, he is very settled and loves playing here.
"We are going to have to work hard to keep him but hopefully we will have a positive conversation with him at the right time."
With Glasgow head coach Gregor Townsend succeeding Vern Cotter in the national job next summer, and New Zealander Dave Rennie taking over at the Warriors, Scottish Rugby chiefs are well advanced in their planning for next season.
Although Edinburgh head coach Alan Solomons is only contracted until the end of this season, Petrie says there is no rush to decide on the 66-year-old South African's future.
"He signed a year's contract and there will be a time when we sit down and look at what goes after that, but the focus at the moment is getting the rugby under way again," Petrie added.
"Alan has the bit between his teeth and is determined as ever to get this club to where he thinks it should be.
"We have recruited well, we have got a pretty good squad and everyone is fully aware we need to kick on now."
Edinburgh have targeted a top-four finish and a place in the Pro12 play-offs as a "minimum" target this season.