Why don't you update the run books? Why don't you modify the alerts/logs to give you more information? Why don't you create the missing run books when you run into undocumented issues?
In our weekly meetings, recurring problems were identified and fixes implemented. No call was considered completed and closed until all relevant documents had been updated as appropriate. At the yearly review the quality of your documentation was as important as your time to respond, time to fix. That is how mission critical on-call work should be handled.