ACT4088 datasheet here
"Output" is from "SW" pin 1.
Voltage appearing on SW when the IC is off depends on load to ground, which you have not stated, and on internal impedance or leakage from Vin to SW when the IC is off.
SW leakage is given on datasheet page 3 in table as 1 uA typical, 10 uA max.
If you were using a 12V supply then 1 uA "looks like" a resistor of
R = V/I = 12V / 1 uA = 12 megohm resistance.
With infinite SW load V_SW could be 12V.
With a 10 mOhm meter - about 5V.
To get ~= 0.8V then Rload on sw could be about
12 Meg x 0.8V/(12-0.8V) ~~~= 860 kOhm.
With 10 uA leakage it could be about 100k.
In fig 2 page 8 the example cct has a feedback divider of about 60 k total and a reverse biased Schottky diode with Ireverse of Murphy_knows kOhms.
ie Without seeing the rest of your circuit and the test conditions it's not possible to be certain, but it sounds like you are seeing about what you'd expect to.
EN pin switching.
According to the datasheet, the IC is turned off when the enable pin is grounded and on when EN is floating OR when it is above 1.24V when sourcing 2 uA. That implies that the effective on resistance of the switch needs to be less than
R = V/I = 1.24/2 uA =~ 600 K.
It seems nearly certain that that is not your problem.
But, any NMOS FET ever made is liable to have an Rdson of << 600k if you choose to use one. Even though it probably won't help - see above.
If you really care about the 0.8V and want it lower AND if the cause is as above, then adding eg a 10 K load should reduce it to about 0.1V, at the loss of Vout/10k when operating.
To get below that you may wish to add a clamp transistor on Vout also driven by the jack. Add a small series resistor to allow it to survive switchover mis-timings :-).
No guarantees :-).
R8 + R7 form a load of about 60k so the 10 uA max leakage should give
V = I.R = 0.6V
This is similar to the result I obtained above for maximum leakage.
Reverse leakage on D3 would reduce this voltage but is probably very small.
The difference between using an MOSFET and a wire shunt is puzzling. The effects of instead using a say 1 Ohm or 0.1 Ohm resistor have not been advised.