I feel your pain. I used to have this problem on a product I worked on and that was with just 2 prod servers, although I did have to jump through 2 intermediate servers to reach the prod server so was a real pain.
Since then we've started to use
Splunk which consumes all the application logs into a single searchable location. Now instead of having to log into prod01, prod02, prod03, etc to find log entries I can simply query Splunk with "host=prod0*". It's much more convenient.
As a secondary bonus, we reconfigured the application to publish its log messages directly to Splunk over TCP which means that the application servers do not have log files containing sensitive information on them. This is a win for data security.