I'm sorry about Fife. What is your new rescue dog's name? Try draining some of energy before leave the house with either a walk, some running around in the yard, or a play session. Give him a little comforting and calming massage.
Try leaving a radio or television on for distraction. I bought a CD of relaxing music called "Through a Dog's Ear" this week and it put my boys to sleep.
It's available at Borders and Amazon.
Try teaching your new dog key words and phrases to associate with your departure, like "I'll be back." Do some training where you depart for only a couple minutes the first time and return. Gradually increase the amount of time you are gone before returning. Be causal, happy, and confident on departure and return. Make him think it is no big deal.
Make sure your new boy has plenty of safe toys and maybe some hidden treats while you are gone.
Hope some of these ideas help,