TI’s application note (PDF!): Single-cell battery discharge characteristics using the TPS61070 boost converter
This application report presents practical single-cell battery discharge characteristics in real-world application, and primarily focuses on the varying internal impedance of a battery and how that affects the battery terminal voltage.
App note mainly tries to convince the reader that the TPS61070 will start up even if the single cell is mostly depleted. A bunch of useful data there.
So TPS61070 should work if you have a manual off-on switch. For a software switch, just don’t switch it off! Quiescent current is 19uA. If you’re not happy with that, switch to say TPS61097, quiescent current is 5uA but it has a lower output current. Much better than MCP1640 but the latter has a slightly lower input voltage minimum…