Try resetting the router with the PC off first.
Okay, let's see if I have everything correct:
1. The PC goes into hibernation/standby and the wireless connection is crippled or totally inoperable when it resumes.
2. The wireless card in your PC is installed into a PCI slot, just as a modem would be.
Odd. Usually it is a power issue that causes this, wherein the PC turns off the wireless card, and then when the WL gets power again the driver isn't able to get the connection up again.
Okay, so this may be more serious that that. Still, no prob!
What brand of WL card is in the PC? If you are unsure, download and install Everest hardware detector and go to the Network icon.
That will show you what network hardware you are running.
Next, go to the cardmaker's homepage and check for newer drivers.
Finally, if there are no newer drivers, send 'em an email and ask them if this is a Known Issue. Sometimes components get sent out with faulty drivers, and this seems likely in your case.
If not, and resetting the router/access point doesn't fix it, you may have to leave the PC running in standard mode all the time.
Really, that's no biggy...just turn off the monitor until the desktop is needed.