How can I power a device that requires 5-7V and 2.5-6 mA with what I have?
If your first device (the "supply") can only provide 0.1mA then you will need an external supply. 

5V@6mA = 30mW

6.23V@0.1mA = 0.623mW 

30mW > 0.623mW

Energy is conserved. 



Edit: You have added that the power source is intended to operate a cooling fan. In that case, it seems much more likely to be 100mA (0.1A) than 0.1mA. There is still potentially an issue with the ground and also the possibility it may be PWM'd or something like that (since the voltage is non-standard for a fan).

An external 5V supply really seems to me like the safest way to go unless you really need to avoid that.