I personally wouldn't be comfortable putting a new fish into water a previous fish passed away in, but at the same time I can understand not wanting to do a 100% water change if your tank is fairly large, cycled or in the midst of cycling. I know with our large tank we'd do a large water change if a fish passed away, but there were multiple fish in the tank so a full clean-out wasn't possible and the tank had a very stable cycle that we didn't want to disrupt.
If I may ask: how big is your tank? Is it heated/filtered? It's quite possible the little one was sick when you bought him and just couldn't cope with the change in lifestyle.
I'm sorry for your loss :(