If you are talking to different ones, then I would suggest getting rid of the pool altogether and just do individual requests that close the connection after the usage.
However, if it is the same website, then a pool is recommended to avoid reopening the same connections over and over again.
The other thing to consider is to actually reduce the number of max_connections. Having a very high limit actually makes things worse in many cases. For example, if the issue is that the upstream website cannot keep up with the load, opening more connections will make everything worse. This does not rule out the other advices in this thread though. It may be all of those factors combined.
