Can I switch the wiring to the gateway to coax?

Not for internet access (coax is for tv only).  How would you use it anyhow for internet? 

Do they also have tv service?  Why was the gateway originally installed where it was in the first place if they don't?    You may want to consider paying att to move the gateway (i.e., it's wall/street connection) to the more preferred location ($99 I think). 

@Johnstwalter 

Not sure I’m deciphering your question correctly, but are you trying to relocate the gateway to improve wifi signal?

in any case, I suggest not using coax for anything.  Att does sell a mesh device that can help boost WiFi coverage.  Look for Airties.  

I guess I was under the impression that you could use coax for internet as well, ( I have used it for other service in the past). They have TV as well. I understand your solution though, and 99 dollars isn’t too bad.

So just to be clear in light of what @gr8sho asked, you are interested in using a wired solution as opposed to a wireless solution?

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@Johnstwalter wrote:

I guess I was under the impression that you could use coax for internet as well, ( I have used it for other service in the past). They have TV as well. I understand your solution though, and 99 dollars isn’t too bad.

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Yeah, Cable uses coax but only to bring the service into your residence.  Once it hits the modem it is converted to Ethernet, just as you see it there now.

If the use of WiFi is not overly demanding and not relying on WiFi Calling feature, using the mesh devices are okay.  But if you need a more robust WiFi solution, I'd look to get better hardware from a reputable networking company.