question about international driving license

If you have an EU driving license and want to rent or drive a car in another EU country, are you allowed to do so with an international driving license while keeping your original license in your home?

The reason why I'm asking is that I just found out that within the EU I have to exchange my driving license from my home country to the country I moved to. Where I'm right now, I don't have an appointment till the 10th October and it will then take an additional 30 days after that to receive it, so I'm a bit paranoid of losing my current license. (losing as in dropping it, not the police taking it)

I thought it would be better to keep my current license safe in my apartment while driving with the international one, especially when I'm travelling, although I mostly travel to Turkey and Greece.