I have a few ideas, but not sure if any have been mentioned before.
The first one I would love to see is interviews with the board before getting a job.
I think you should be able to have an interview in kind of like a board interaction style meeting in which you outlay plans or at least long term objectives, tell them what kind of football you'll be bringing to the club and other questions I haven't thought of yet.
I think it would add to the sense of achievement when you actually get a job rather than just holidaying and waiting. Maybe even be able to ask the board questions some how so you know what they are expecting in more detail/more long term.
In real life you hear a lot about manager relationships with the board/chairman, it would be good to get it more involved within the game to.
Another idea is maybe adding in a press interview after you have resigned or been sacked from a job.
Outlining why you quit, or your view on why you were sacked with the aim of bigging yourself up for your next job.
You often hear about managers being sacked for no apparent reason (Lee Clarke for one), would be good to be able to come and have a moan.
Also think that the way you receive training reports needs to go into more detail.
It would be good to know exactly how players are training week by week so you can give them a rollocking rather than relying on a generic coach report to show you a sliding bar or just have the players say they are either happy or unhappy with training levels.
Not sure how this could be done, but I think it's an area that could be improved.