Applying a voltage at the output of a linear regulator
I'm not familiar with the regulator but a common trick on PSU design is to feed back from the output to the input using a diode. This way the input can only be one diode-drop below the output. Using a Schottky diode will minimise the voltage drop. You'll have to decide if this is enough.



simulate this circuit &ndash; Schematic created using CircuitLab
Figure 1. The scheme - if the +6V is available on the header - would be simple.
Judging by @Rohat's comment this is risky.
The obvious solution then would be to apply +5 V to the +6 V input too. This eliminates the problem if that pin is easily accessible.