It should possible to go to your existing registrar and change the nameservers to point to GoDaddy before you start the transfer and get the site live.
Once the new site is live then have a look at doing the transfer to GoDaddy, that sounds like the safest way to me. During the process of transferring from one registrar to GoDaddy the nameservers should remain the same.
-----However if that doesn't sound like a good idea-------
In a perfect world transferring your domain should keep your DNS settings the same resulting in no downtime. So if you was to transfer first, then point DNS later you should still be able to avoid downtime. Depends if your old host terminate the service or if GoDaddy automatically change the settings.
And as always with either option when you change the DNS over there will be that propagation for the domain so for the 24 hours to follow people will either get the new or old website.