How to pick a rechargeable battery

I assume that you mean that the current is about 1/4 of the original current.

Phone batteries will almost invariably be Lithium Ion (LiIon).
LiIon are normally charged at a maximum of "C" = the rate in mA numerically equal to the capacity in mAh.
So, for a 1200 mAh LiIon battery C = 1200 mA.

Assume your battery is 1000 mAh.
Actual values can be scaled if this assumption is wrong.
Normal maximum charge rate = C = 1000 mA.
Solar charge rate = 1000/4 = 250 mAh.
Assume it is a 3.6V nominal battery. This is the voltage of a single LiIon cell.
Max allowed voltage = 4.2V. Charge it with more than 4.2V and it's lifetime will be severely impaired OR the battery will self destruct.

IF the battery voltage when charging is less than 4.2V the 1000 mAh battery can be safely charged at 250 mA with no ill effects.
Once the battery voltage reaches 4.2V on charge the current MIST be controlled to keep Vmax at 4.2V OR the charging must be terminated. IF you continue to charge at too high a rate when Vbat reaches 4.2V so that Vbat keeps rising the battery and perhaps its surrounds and perhaps you will be damaged.

It is not enough to just limit Vmax to 4.2V. If a LiIon battery is connected to 4.2V indefinitely it will be damaged.

In the case of a small solar panel and a LiIon battery three main choices exist.

1. Feed the PV panel voltage into a LiIon charger and use that to charge the battery. A number of camera chargers have 12 VDC inputs and can be charged from an external voltage source - with the charger electronics looking after charging complexities. or

2. Charge to 4.2V and then terminate charging completely - do NOT "float" at 4.2V. or

3. Build your own LiIon charger using one of the many purpose built ICs made to do just that OR 'roll your own' with an opamp, voltage reference, control switch (MOSFET usually) and more.
   Option 1 is easiest.

I think that a very good option is to take advantage of the PIC uC that you already have in your design and its analog (ADC) input ports for monitoring the batteries voltage and to drive the switching between those batteries using MOSFETs instead of BJT transistors. Using the uC you should not need the two comparators that you show in the picture and you will have a more intelligent charging/monitoring battery circuit that could lead to improve the batteries duration. This uC could stay in sleep mode (or low power mode as second option) all the time using interrupts to read the ADC channels. You should run the uC from its internal RC oscillator operating at the lowest frequency as possible to decrease the power consumption.

I also recommend you using Schottky diodes instead of 1N4148 and to put a small bulk capacitor at the output of the power selector circuit (or switching circuit). Those Schottky diodes will drop less voltage than the 4148.

If you decide to use the uC for the battery voltage level monitoring, try to use the internal fixed voltage thresholds that the uC provides for the internal comparators, because if you don't do that you could have some problems reading wrong analog values because those batteries are used to power the uC too (you will have ratiometric voltages).

I hope this help.