ICANN 60 days rule.
Well it's not apply in your case, since the domain is already register 5 year ago (not 60 days ago), and I've assume you've not change any detail from last 60 days.
Domain transfer is really easy nowadays, it doesn't require time at all, you can transfer any domain name in only one day, or max two days. Also you can change anything you want with valid details.
First you need authcode, which you already got (your company might have first disable the protection and then request for authcode)
Now the procedure are different from one registrar to another registrar, because if you're in same registrar then ask for their help, they might transfer the ownership within their network so easily, but if you want to transfer your domain name from one registrar company to another, then you need some extra work.
Make sure your domain is not protected with who is protection, so disable the privacy of your domain first, and make sure it show valid email address.
Go to the any registrar and follow the procedure to transfer, there make sure you enter the valid new details. Here you're fully allowed to change anything you want including name, email, business address, phone etc.
That new domain registrar will look for who is information and send the confirmation link to the old email address(which might accessible to only your company which you've hired in past, so talk with them and say please confirm the approval).
Your new regisrar now have power to control your domain name, but first it should be release from your old domain name. Again to release the domain name from your old registrar quickly there are some procedure, so simple follow their FAQ about how to release the domain name or approve the release of domain name quickly. For godaddy you can reference this article. Once you do that step, simply ask your new regisrar to speed up the process, they will do it, since you're new customer for them.
If you follow 4th step cleverly you can easily transfer any domain name to any registrar within one or two day, otherwise your old domain registrar will wait for 5 to 7 days, and if they don't receive any complaint about that transfer, then they will approve the transfer of domain name automatically.