The decision means initial works scheduled for February can now include other aspects of the wider project.
Hamilton mayor Paula Southgate says she was delighted elected members unanimously endorsed the staff proposal.
The shared entry precinct and a safe crossing point for visitors between the two facilities supports sharing Hamilton's natural heritage with its community.
Combining this work with an urban upgrade of this section of Baverstock Rd also benefits local residents through lower speeds, safer roads and a better urban environment.
"The overall project will deliver new footpaths and cycleways and improve public transport so there are some real wins," Southgate said.