Hague ran for the male co-leadership of the party last year, but was defeated by Shaw, a relative newcomer.
The West Coast-based MP was respected on both sides of the House, and would have been a certain minister in a potential Labour-Greens Government.
Among his biggest victories were lobbying for a culture change at ACC in 2012, and playing a leading role in the legalisation of same-sex marriage the same year.
Turei said his successor, Coates, will bring extensive experience in social justice, climate change and Pacific issues.
"He will be an important new voice in Parliament and we look forward to working with him."