I'm not sure what you mean by "stale". Is it bitter? That's the problem I usually start to taste before I know an in depth cleaning is necessary.
Make sure you double up the pipe cleaners for the shank and the bowl as they are a little more open than most briars.
How many bowls is a "few"?
Have you ever taken a cigar and smoked it half way and let it go out then relit it up and its not as enjoyable as it was the first light up? That stale old smoke taste is what my pipe seems to have. I think I've smoked about 10 bowls now giver or take a bowl. I do get a bitter makes me take the pipe out and spit a bit taste but that's normally when some of my saliva gets into the pipes mouth piece and then gets into my mouth. My cob seems to have enough room for two pipe cleaners so I'll start doing that now.
Doubling up the pipe cleaners will likely help. Make sure you use a bit of alcohol of some kind, too. Pay extra attention to the stem (or mouthpiece). Also, make sure to let the pipe dry out completely before the next smoke after cleaning.
If none of that works, I'm not sure what to tell you. I've rarely had the problem and whne I have, a good cleaning always works.
Good luck!