Novice question about nameservers.

I am using a web hosting company for our businesses email and web site with
our own domain. Our hosting company has been down for two days and my boss
wants to know if there is a way to avoid downtime in this situation.
I have an idea which may be wildly wrong but here it is; we set up another
hosting account (acc B) with a different host to work alongside our current
account (acc A).
I then set our domains name servers as follows:

1. Acc A primary
2. Acc A secondary
3. Acc B primary
4. Acc B secondary

I think this will mean that it Acc A becomes unavailable the mails will
automatically go to Acc B.
I am aware that we will need to run 2 mail accounts in outlook to check both
hosts.

Does this all make sense? If not are there any other ways of achieving the
desired result?

Thanks in advance for your time helping with this, it all seems to make
sense to me but perhaps I am missing something...