The output (red) could be connected to the Enable pin of a Boost converter IC (LM2623) (blue line is the battery voltage)
Vstart=1.37V
Vstop=0.86V
PNP transistor: 2N2907
NPN transistor: 2N2222
From the document "MCP16251 ONE-CELL BOOST
CONVERTER WITH EXTERNAL
UVLO CIRCUIT"
Document Id: DS50002561A