To address your concern about bypassing the DC/DC converter when the solar panel output is below 6V, you can consider incorporating a simple voltage-triggered switch using a Zener diode and a MOSFET.

Here's a conceptual explanation of how you might modify your circuit:

Zener Diode Setup:

Connect a Zener diode in reverse bias across the solar panel output.
Choose a Zener diode with a voltage rating slightly above 5V (e.g., 5.1V or 6.2V).
When the solar panel voltage is below the Zener voltage, the diode remains off.
MOSFET Switch:

Connect a MOSFET in series with the output of the DC/DC converter.
The gate of the MOSFET can be driven by the voltage across the Zener diode.
When the Zener diode conducts (solar panel voltage is above its threshold), it turns on the MOSFET, allowing the DC/DC converter to function.
When the Zener diode is off (solar panel voltage is below its threshold), the MOSFET is off, effectively bypassing the DC/DC converter.
Implementation:

Test and adjust component values based on your specific requirements.
Ensure that the Zener diode's breakdown voltage and power rating are suitable for your application.
Choose a MOSFET with low threshold voltage and appropriate current-handling capabilities.
Here's a link to a detailed article on the 7805 voltage regulator IC, which might provide additional insights into voltage regulation: https://www.jotrin.com/technology/details/7805-voltage-regulator-ic-pinout-circuit-datasheet-uses

Just add a voltage doubler, then 2-3 diodes in serial, then add a 7805 Voltage Regulator IC (or keep your existing stepdown module)

outputs 5V, input can be 7,5v - 12v

https://www.electronicshub.org/understanding-7805-ic-voltage-regulator/

7805 has a drop out is at least 2V to maintain regulation. i.e. need at least 7V input. So 7805 efficiency is always below 71%.

At max 8V input, your doubler produces 16v which will be stepped down all the way to 5V.  So efficiency will be at most 31%.

Yeah, you need a Buck-Boost.

Did you consider using a straight-up DC-DC Buck? When the voltage drops below the regulation point(i.e., 5V) the FET just shorts so it's a very efficient way of doing what you need vs a LDO style regulator. Something along this line.

http://www.ti.com/document-viewer/TPS62120/datasheet

Of course, you can make your own buck as well but the control aspects can be annoying so I usually prefer a all-in-one IC. 

This look realy interesting although 75mA might be not enough, so probaply more like TPS62125.

Looks like a plan :)

Also, I'm assuming you don't need regulation at 5V when the panel dips below 5V; which is why I don't recommend a Buck-Boost. They are usually inefficient and when it comes to small solar panels, every mW counts. 

Also adding to @fdufnews answer. Usually MPPT + Regulation is dealt with as two stages to maximize solar panel throughput. Typically MPPT can't be achieved without screwing up regulation, unless you have a clever way of sinking the exact amount of current. So the MPPT circuit output voltage swings depending on how much power is being generated; for regulation you have a second stage which keeps this at a constant voltage. 

All depends on what you intend to do with the solar panel. If it's just running some electronics and you don't intend to use it for battery charging; MPPT is most likely overkill. 

Maybe you can open your powerbank and look at the circuit inside.

For example, many LiIon charger are using a TP4056. This circuit operates from 4V to 8V. It goes in stand-by if input voltage drops to within 30mV of the battery voltage. Providing your solar panel never output a voltage higher than 8V, in that case you don't need any DC/DC.

But the major problem is that the powerbank expects the voltage source delivering some current (usually 1A or so) and doesn't limit the current under that value, this will tend to overcharge the solar panel whose voltage will drop making the charger enter stand-by then voltage will increase charger will restart,  .......

Solar panels don't ouput constant power as phone chargers do. So a solar charger needs to be a little more intelligent. Hence the MPPT converters that maximise power transfert from a solar panel

This idea is very good. Also it pointet me to another problem: when the volatge is this low, also the current is low so charging the battery might not work at all...

I will try to open it up and look what is inside. As the DC/DC is shutting down it might prevent the battery from getting damaged.

How about a buck-boost converter? 

I dont know if they will work well with solar panels as their current is pretty low (100mA at best). According to this (https://electronics.stackexchange.com/questions/204937/what-would-happen-if-you-connect-a-solar-panel-to-a-buck-boost-converter) it might not work well with high loads.

My full ideas is to plugin a USB powerbank to the 5V output. Unfortunetly I don't have a detailed datasheet on what input range is already accepted by the powerbank.

May be connect the input to a 5V buck converter. Use it to extend your battery bank range.

Most switchers that uses NMOS have a max duty cycle of 90 to 95% which means that they regulate down to 5.5V input. The output would be max duty cycle x input voltage below that point.  This isn't too bad actually. You just have to make sure it is not an old buck regulator that uses BJT for switching. 

BTW the Chinese module (in green) I use here uses a MP2307 chip. It has max duty cycle of 90-95%.

https://1.bp.blogspot.com/-GjQGAHU35VI/W2IDKtBBwDI/AAAAAAAAB-A/TM4Mv9eN2Mc2sO2fZCEReO37ab8RBp7vACLcBGAs/s640/2628971466880202134.jpg

LM3485 can operate in 'dropout' mode.  At 5V or below, the PMOS would be turned on fully with very little drop between inut and output. This helps to extend the input range a bit (5.5V to may be 4.5V) for charging the power bank.

https://github.com/FPGA-Computer/UPS/blob/master/Buck/12V%20-%205V%20Buck%20(1A)%20-%20LM3485%20v2.pdf