Solomon has led Essendon to one win from nine matches since Brad Scott was sacked in late May, but was widely tipped to enter the race to become the club's permanent coach.
At one stage he was even considered the favourite.
But the 46-year-old officially pulled out on Wednesday in a decision the club said "reflects Solomon's long-standing commitment to doing what is right for the Essendon football club".
"I've been incredibly grateful for the opportunity to give back to the football club that has given me so much," Solomon said in a statement.
"I'm proud of this playing group. While the results haven't reflected where we want to be, I've seen the growth of these players every day, and I honestly believe our members and supporters have every reason to be excited about what's ahead.
"They're made of the right stuff, which gives us great hope for the future."
Solomon's decision means there are now four names left in the race - Hird, Fremantle assistant Jaymie Graham, Swans assistant Mark McVeigh and Adelaide coaching director Murray Davis.
Hird is an Essendon legend, having captained the club during his decorated 253-game career.
But his coaching stint at Essendon from 2011-2015 was marred by the club's supplements scandal, with Hird suspended for the 2014 campaign as punishment.
Essendon fans are heavily divided about Hird's potential return and it's that division which could ultimately cost him the job.
Essendon chief executive Tim Roberts paid tribute to Solomon's efforts in helping the club during a tough period.
"Anyone who has worked with 'Solly' understands the person he is," Roberts said.
"His leadership, integrity and care for our players have been outstanding and we are incredibly grateful for the way he has led our football program through an important period.
"He has served this club with distinction as our senior coach and has made a lasting impact on the players and staff around him."
Essendon (2-18) sit last on the ladder and have endured plenty of tumult both on and off the field.
Assistant coach Cam Roberts, who has been in charge of the forward line for the past two years, announced this week he will depart at season's end.
He joins Scott, football boss Daniel McPherson and Indigenous player development manager Travis Varcoe as confirmed departures following a season from hell.
Essendon will be back in action on Saturday when they take on Geelong at GMHBA Stadium.
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