Making a counter that goes from 1 to 6
A decade counter isn't particularly suited for this type of task. The 74LS390N does have a "CLR" pin, which makes the output immediately back to 0000, but this doesn't do much good, as you want 0001. You could make some clever tricks with additional logic, but that wouldn't be easy, and it isn't a flexible option in general.

Tasks like this are generally done with a microcontroller. It would replace the 555, the '390 and the associated circuitry around them, and gives you flexibility as an added bonus.