How did airline and or mail pilots avoid hazardous weather especially at night before airborne weather radar systems were in use?
They looked out the window!
If you are talking about a developing thunderstorm, especially during the day, they can be quite easy to see in front of you provided decent visibility. Here is some good footage of what it looks like to fly around a towering cell, you will note its fairly easy to see. At night, lightning, even in the clouds tends to be a prominent effect and is a good indicator of "dont fly there". Here is a good picture of that effect (as well as a little story about flying around it).
Here is an interesting report on thunderstorms and aircrafts from 1948 that may have some interesting info for you.
I´ll second John K´s recommendation of Fate is the Hunter for a first hand account of flying in the 30s and 40s. The style can be a bit too laden in floritures, trying to lend a certain mystique to flying that still existed when it was published, but the content is sound and if you want a glimpse into the aviation world of that era, it is a great read.
To summarize some of the main points:
- Reports of winds aloft were reports of the local conditions relayed by radio from aircrews flying in the area of operations and served to inform following pilots.
- Experience handed down orally by more senior airmen that were familiar with a particular route.
- The poorly understood impact of icing on an airframe made weather seem like less of a danger than it really was. Since CVRs and data recorders were still far in the future, many weather-related accidents were unsolved.
Read the book "Fate Is The Hunter" by Ernie Gann, who was flying DC-2s and 3s for American. One of the all time great aviation books.
In the 30s, they just blundered right through thunderstorms if they couldn't be visually avoided because they were embedded in cloud or it was at night, and hung on for dear life.