Possibly the appropriate network adapter is not initialising when the computer returns from hibernation. Wait... did I say "Hibernation"??? Plain shutdown leaves the computer in hibernation rather fully shutting down when you use the default "fast start-up" startup mode. (Restart always does a full boot from scratch when the computer restarts, by default, shutdown/power on does not)
On the next startup after a plain shutdown, the computer can simply return from hibernation rather than perfoming all that goes with a full boot up (this is a lot quicker to do). If drivers do not correctly reinitialise devices after the hibernation, strange effects can occur. Try disabling fast start-up to see if this resolves the issue.
To turn off "fast start-up"
- Go to "Power Options". (search for it from the start panel)
- Select "Choose what the power buttons do" (a link halfway down the left hand side).
- In the bottom section ("Shut-down settings"), deselect "Turn on fast start-up". If this box is greyed out, you may need to click the "Change settings that are currently unavailable" near the top of the page first.
- Click "Save changes"
If the setting does not help your issue, you can revert the change simply by going back to the same place and ticking the box again and saving the change..