Yes you certainly can catch another cold just after getting over your current cold virus.
You could be catching another strain of the cold virus that is lingering in the air or be catching the cold virus from someone else that you haven't had before.
There can be more than just one type of cold virus so it's entirely possible to get over the cold virus and feel better and then a day or so later or even weeks later you could catch another cold and start feeling bad again.
Same goes with the flu virus as it's entirely possible to get over the flu and then a week later catch the flu again due to different strains of the virus.
I've gotten 3 colds in a year last year when I normally get only 1 cold or 2 colds at most in a year.