Rather than scheduling a one-time job, schedule a recurring job.
Schedule the job to run on an hourly interval (every hour). As part of the finishing phase of your job, cancel this hourly schedule and replace it with another similar hourly schedule where the first execution is set to be a short period (let's say 5 minutes) from the finish of the job.
This works in a very similar way to using a "one off" schedule (as per your existing implementation) - in both of these implementations the job is rescheduled in the finish phase, but by using a recurring schedule you have the added benefit that if for any reason the job does not execute, the platform will attempt to run it again an hour later, and every hour until it succeeds.
Note that we don't know why the job may fail to execute - but we're assuming that it relates to platform maintenance. Chaining one-off scheduled jobs together relies on the successful start and completion of each job for the integrity of the chain, whereas using a recurring scheduled job provides "auto-resume" behaviour regardless of the successful start / completion of an individual job.
Example process flow:
(1) at 12:00 we schedule a job to run every every hour, at 5 minutes
past the hour: 12:05,13:05,14:05...etc...
(2) at 12:05 the batch manager job is started according to the hourly
schedule, and this checks your custom batch job object records to see
if there is any work currently running or waiting.
It finds that there are no jobs running but there is a job waiting:
"Foo". The batch manager therefore starts the batch process for Foo.
(3) at 13:05 the batch manager job is started according to the hourly
schedule.
On this occasion it finds that job Foo is in progress and so quits
taking no action.
(4) at 13:35 job Foo finishes.
In the finish phase, the existing hourly scheduled job is cancelled,
and another new hourly job is scheduled, this time to run at 40
minutes past the hour: 13:40, 14:40, 15:40...etc…
(5) at 13:40 the batch manager job is due to start according to the
hourly schedule, but this fails (we assume because of platform
maintenance)
(6) at 14:40 the batch manager job is started according to the hourly
schedule.
It finds that there are no jobs running but there is a job waiting: "Bar". The batch manager therefore starts the batch process for Bar.
etc.
Best Answer
I'm afraid that you would need to have the Debug Logs running on the user who scheduled the actual batch job, as well as informative debug code (e.g. System.debug) within your Scheduled Apex class. If you to Setup | Monitor | Logs, you can add the user record which should be monitored. Resulting in different debug logs.
Note: only 20 logs can be created. So in case you have batch job scheduled during midnight, set up the user for logging right before leaving the office (hereby ensuring the user running the batch will not browser anymore and hereby not consume any of the 20 logs that can be created). Check again in the morning on the same place, you should be able to find back the logs from the batch.