Battery auto backup for GSM
Essentially you want to make a UPS - Uninterruptible Power Supply for your module. This can't be solved by means of simple "comparators and MOSFETs", it would also need to manage the charge of the cell, and provide smooth power bypass. 

You already have a solution, posted in your previous questions. It is one of PMIC from TI, BQ24xxx/BQ25xxx ICs. You just need to override internal defaults (watchdog timer, etc) using I2C interface. 

If you don't have a MCU with I2C port, you need to look for a stand-alone circuit, something like this one, or similar, although it will be difficult to determine if these kind of designs will provide all functions you need. 

Another alternative is to find a proper powerbank module which allows internal battery charging and external power delivery at the same time, which would be also a challenge to determine without buying samples first.

If designing the circuit from scratch, you might want to look into "Prioritized PowerPath Controllers" similar to this one, LTC4417.