Powering a Tiny Arduino Project
The old circuit will not work as you expect. While the ATtiny85 can run happily with 2.7 to 5.5V in this setup it expects 5V. I assume this is due to the high CPU clock setting. To run stable at a high CPU clock it requires 4.5V.

The VIN pin is connected to a linear regulator that has a 2V drop-out. Thus the supplied voltage has to be at least 2V about the target voltage. That is why the shopping page states a 5V (direct) or 7-35V (regulated at VIN) power supply.

Your LiPo charger already has a boost converter built in that boosts the battery voltage (3.0 - 4.2V) to 5V. No need to use a separate DC-DC boost converter here.  

Just connect OUT+ to 5V and you are done.