External cellphone battery - to keep on, or only use when phone battery low?
Downvoter may be reflecting that this question is OT for the site.

What happens is complex and would need analysis for specific cases, but I'd expect to get somewhat more capacity by discharging to 40% (or less) and then recharging rather than floating at full charge. Because:

LiIon battery charges at CC (constant current) max allowed current to about 70%-80% of charge then enters a CV - constant voltage decreasing current mode. 

When operating in CC at highish load the boost converter is probably more efficient than in CV mode at lower load. 

In top-up from start mode the converter will be being cycled on and off (probably) and will be 0% efficient when standing by (but at low load. 

As well as the converter, the battery itself charges more efficiently at lower % capacity levels. A LiIon battery is close to 100% CURRENT efficient when charging but in the top 0% or so voltage rises faster than charge % as cell impedance rises and energy efficiency drops.