Charging a 4.4v lithium polymer battery with 5v?
You could make a custom charger BUT I'd strongly recommend against it.

The energy difference between 4.2 &amp; 4.4V will be small and you will get useful whole of life capacity improvement by using 4.2V end of charge.

You can test this by 


CAREFULLY manually charging to 4.4V with a power supply set to 4.4V and current limited to Imax, 
then letting the current drop in CV (=4.4V mode) to say Imax/2 
THEN discharging to 4.2V - ideally at constant current but a fixed R would do. 


See how long that takes, which will give you an idea of the energy / mAh in the 4.2 - 4.4 V area. It will not be vast compared to whole battery capacity. 

The TP4056 should do a good job