Move Mobile Browser Chats to SMS or a new browser window

Most of our customers visit our website via mobile browser. When the Zendesk chat/messaging widget is selected, it takes up the entire screen. Customers accidentally leave the chat if they want to reference the website again.

We used a marketing company for a while, and they redirected customer conversations to SMS. We thought that was a nice solution to our problem, but we no longer want to work with them, and we'd rather keep everything in the agent workspace in Zendesk. 

Is there a way to open the Zendesk chat widget in a new browser tab so the site and the page the customer is chatting from are still accessible to them?

Is there another solution to this issue we haven't considered? I know switching from chat to messaging would allow the customer always to visit a past conversation in the widget, but we wanted to know what else is possible.