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Picking up all the Green votes seems incredibly unlikely though, they're not a directly analogous party to Labour. The Green vote increased between 2017 and 2019 so they were clearly looking for something more than left leaning policies.
Plus, doing so without losing other support would be incredibly difficult. Trident is the obvious example: would binning that to win Green votes win more votes than the policy lost?
Does he really need to gain ground with the "left"? To be brutal there aren't really many other options for them.
The Tories have a huge boost at the moment with the vaccine program going well. Labour were ahead in the polls a couple of months ago.
Anecdotally Starmer seems to be making a positive impact on a lot of those voters that Labour had lost but years away from a general election during a national crisis probably isn't a time to judge with any certainty.