Of course I agree we are only human, and we've all made some mistake like yours at some point.
But I believe adding any fallback option is just shifting the problem. At some point you have to click "send" for real where there is no going back, so why not just at the current "submit"?
Also, adding some way to resolve scenarios like yours does generally also open up other problems.
Ideas (mostly from comments/the question) - I've shown the drawbacks to the ideas mostly, as the benefits are already obvious:
Show company contact info
The likelihood of spam potential means I doubt this would ever be implemented.
The application is not sent for 5 mins after hitting "submit" and can be edited.
The Question and Answer edit grace period on the sites is just so trivial typo fixes (etc) don't have pointless historical logs.
However, with submitting an application, a grace period is only extending/delaying the problem.
Why hit submit and then think "have I done everything", rather than doing it before hitting submit?
What if you think of something 20 mins later? Should we have a grace grace period? Or just have it set to 30 mins instead of 5 mins?
I'm not being inconsiderate towards people making mistakes, I just don't think this adds any benefits.
IMO, you just need to check it thoroughly before pressing submit. And while clicking submit can clear the mind as some pressure is released, and so we "think of something else", a simple (and arguably better) alternative is: Before submitting, go away from the application for 10 mins or so, do something else, and then come back and have a final check.
Once (and only after) an application is submitted you can send something else.
This isn't too bad, but it opens up for nuisance emails, with or without an expiration time:
"Did you read my application yet", or "Here's some more info"
It also shows we "made a mistake", and without it users might be more "thorough" in their final checks.
Additional notes + blocking options
Expanding on (3), allow additional notes after application is sent, but also allow companies to choose to block or receive additional notes "per application" and/or "per user". Allowing each company to tailor this to their policies/requirements.
So a company can choose to block notes for this application, then the notes function is not even present - "this company does not allow additional notes on this application" (informed before submitting).
And if a single user is sending too much junk, they can just block that user.
This might be a great option, but it also seems like a chunky system - might be too much dev time.
For any of those options, dev work might not be a good ROI, especially as the company doesn't really benefit any at all, not over you just making sure the application is completed accurately.
Could back fire on you
While I appreciate we all make mistakes, it's probably not in your interest to have a "backup" to rely on. With a back up option users might not check their application thoroughly knowing they can send something else, or have that 5 mins grace to check it again after hitting submit.
Also, with the "additional notes", being able to send through more than one item means the company has to tie up "multiple" emails or documents for a single person - which could go against them when everyone else has managed to complete the application form successfully (a single document).
Yes, some companies will be fine with this too (and some will also allow additional notes to see if you make mistakes under pressure..etc).
However, considering the dev work, users sending "corrections", and companies having multiple documents, and all just to cater for making mistakes, doesn't seem worthwhile a good procedure to allow.
I expect some downvotes, as people will be against this because they make mistakes.
However, while I know it's frustrating when we make a mistake like this, I do myself, in essence we should really be making sure everything is in order before submitting.
Especially knowing it's so critical.
It's not unlike an interview, or exam, whereby you get that one chance and any thoughts afterwards is simply too late.
And we should be showing companies we are professional and can at the least fill out an application form, without needing to send through "corrections" - especially as we're not even in front of them or in an interview, and are at home/work etc able to spend time checking it.