with primary cells you would power the ESP via lowdrop/lowpower linear regulator. That makes max. 40µA (esp, flash, ldo) in standby. When the ESP powers up it powers the sensot, takes a reading, collects and/or transmits it and goes back to deep sleep. Either run until it doesn't power up any longer or just take a vcc reading every wake-up and send an alarm and set "sleep forewer" if under threshold.
with optimum LiIon (3.2V LiFePO4) you power the esp directly from the cell. Charging can be done via e.g. USB (smartphone charger), solar cell, etc and LM317 configured for 3.5V. LiFePO's are very robust (charging wise) and have almost constant voltage decharging. Battery monitoring as above. Standby is a little lower since ldo is not involved.
with standard LiIon (3.6V Mn,Co, etc) you go for one of the USB charge/protect boards (0.45$US, could also be charged via solar) and the rest like primary cell. The standby is a little higher, caused by the protection circuit.
It may be that the solution with the ATtiny has lower standby but the partcount is signifficantly higher and parts are more costly. Also from my point of view, it is less elegant ...
It is possible to further reduce the standby, but only if the "trigger" is external.