Add Party to Call without Putting Other Parties on Hold - Amazon Connect

Connect does not support this natively, and to complete a conference then transfer, the 3rd leg of the call must pickup before additional steps can be taken like join all the legs together.

You can use a transfer flow to conference before dialing the number - note this will still put the caller on hold while setting up the new call leg, but it's only 1-2 seconds, then the caller and agent can both hear the call being dialled.

• Create a Transfer to queue flow
• Add a Hold customer or agent block with status Conference all
• Add a Transfer to phone number block with the number you want to conference in
• Publish the flow
• Create a quick connect for a queue (doesn't matter which one), selecting the new flow
• Add the quick connect to the relevant queue(s) so it appears for the agent

Since you mentioned emergency services, note that there are some limitations called out in the AWS service terms - see section 54.2 under https://aws.amazon.com/service-terms/