Hm, I guess I'm unclear what the initialization gap you're referring to is. The task event you include ("Exit Code: 1 ...") occurs if the Docker container's entrypoint is run and then exits with a non-zero status code. Since at this point the entrypoint has been executed neither Docker nor Nomad have any additional metadata about why the entrypoint exited. It's up to the entrypoint's code (eg redis, your application, etc) to log why it's exiting and those logs are accessible via the nomad logs command, API, and UI until the allocation is garbage collected.
Are you asking that Nomad include somethling like the last line logged in the task event? If so, feel free to open an issue. That seems like an interesting enhancement to pursue.
I'm just having trouble imagining what other data Nomad could provide. I've found myself facing this error a lot too, so if there's a way to improve it: I'd love to!
-schmichael