I frequently wear a pair of Bluetooth headphones paired to a computer. If I want to listen to something on my phone I have to re-pair the device to my phone.
Is it possible, through software or hardware, to have both my phone and computer connected in such a way that I can get audio output from either device to my headset simultaneously.
You need headphones with multipoint functionality. If your current pair can't connect to multiple sources at once already then you may need to buy new ones that can.