Function opposite of blackout delay

[CaptCrash]

I think that this can be done with some programmable logic and a timer.

Create a timer that is on for the periods that you want
Timer 1
set times that you want the device to turn on (In a normal situation the socket is only active a few minutes every day), so 9:00 am to 9:04 am and 5:00 pm to 5:04 pm for example.

Create two programmable logics

Programmable Logic 1
This PL should leave its output low for the delay time specified, then changes the output to high.
Function: Delayed On
Input: Always On
Time: 240s (time you want the socket on for in seconds on a system restart)

Programmable Logic 2
This PL takes the input of PL1, inverts it (starts high then after the delay period goes low), and compares it to the timer 1 function. If either the timer 1 or PL1 (inverted) are high, then it sets its output to be high.
Function: Or
Input 1: Programmable Logic 1 (Invert)
Input 2: Timer 1

Set a socket to be controlled by Programmable Logic 2
Socket 1: Programmable Logic 2

I have worked this through in GHL Control Centre, but not tested the operation on a power cycle of a controller. So it is based on the assumption that the delayed on function will work on a power cycle.
Hope this helps or at least gets you close to what you are looking to acheive.