If logged out from Salesforce or for users, not working with Salesforce, you can make a phone call using the Browser Client (CCP) from Amazon.
It can be launched using the link below
https://YOURAMAZONINSTANCE.awsapps.com/connect/ccp-v2
(you have to replace YOURAMAZONINSTANCE with your Amazon Connect instance name).
To make phone calls, the Browser must be opened all the time.
As an alternative, Mirage offers a Software Phone Client application (Windows / MAC), which can be automatically launched when the operating system starts and is always available. It uses the CCP Browser phone but is a real application (EXE or APP file). You can make phone calls as usual, and you never miss a call because the Browser was closed. Also Amazon Connect chat can be enabled.
AC SoftPhone by Mirage
It can be ordered on the Amazon marketplace.
Online Help and full product description.