Possible to speed up propagation time?

I am in the process of moving a site from one host to another and was curious if there is anyway to speed up the propagation time?

The domain name is registered through GoDaddy. Overall I know they say it can take up to 48 hours, but usually it has been quite fast in the past and for some reason this go around it has been already 10+ hours and the site is still loading on the old host.

If anyone has any tips it'd be much appreciated.