It depends on what you mean by effectively. Could you elaborate on what's not working well with the current set up?

A few suggestions though based on some assumptions;

1. You could create a new team inbox specifically for escalating conversations to the product/technology team. Your support team can then change conversations to those teams when they need help, and they can just go through the queue and then send it back down to support with some action. This would also allow you to report on this, so you could see how many conversations are escalated to those teams and work on ways to reduce that. It can be a helpful way to identify learning opportunities to reduce conversations needing to be escalated to those teams by empowering your support team more.
2. In conjunction with 1., you can automate some manual typing by creating macros. Some more info here: https://www.intercom.com/help/en/articles/5050411-creating-and-managing-macros/li>

If you provide a bit more info about the setup and what you'd like to achieve I can give some more advice