[PJones]

You can get this from most places now:

'Hi, I was just running through the troubleshooting steps with one of your colleagues and I got cut off. He said he was just going to put me on hold to [order new router|arrange engineer|whatever you think is needed after your own troubleshooting]'

This works for me 95% of the time.

[SatvikBeri]

Slightly sleazier version: use a similar trick to get a discount when ordering new service, e.g. "I heard I could get a discount/fee waiver for plan X."

I don't consider this particularly sleazy because you can usually get the discount anyway if you haggle, but it saves a lot of time.

[PJones]

This works even better if you know what the deals are ahead of time, either from searching online or by knowing someone that works there.