Adler had been at the club since moving north from Leverkusen in 2012, and played 117 Bundesliga games for the Red Shorts. His final appearance came in the 3-0 Matchday 27 defeat at Borussia Dortmund when a broken rib ended his season early.
Despite the club preserving their top-flight status with the final-day win over Wolfsburg and maintaining their proud record of being the only Bundesliga founder member never to have been relegated, the 32-year-old has decided not to extend his Hamburg contract.
"It's my decision," Adler, whose current deal expires at the end of June, said in a Facebook post. "I don't have a Plan B, and there are no talks with other teams. I still don't know where I will be after the summer break."
He added: "I don't go holding a grudge, but with a lot of sadness, especially after such an emotional end to the season! Hamburg has become my home and will be again after the end of my sporting career."
"We respect Rene's wish for a sporting change," said HSV sporting director Jens Todt.
"He's been one of the defining faces of HSV for more than five years. We thank him for his achievements in the HSV shirt, and wish him all the best for the new challenges ahead."