By adding a bit of code to "Email (New)", you can set a customer custom field to the user's timezone. Before adding code to your support center theme's contact form, create a customer custom field with key "time_zone" and with the 4 values used below.

gistfile1.html

<!-- Add this right above {{form_end}} --> 

<div class="timezone-div">{{ customer_custom_time_zone }}</div>

gistfile2.html

<!-- Add this script to the very bottom of Email (New) -->

<script>
function getTimeZone() {
  return /\((.*)\)/.exec(new Date().toString())[1];
}

$(function() {
  timezone = getTimeZone();
  if (timezone == "EDT" || timezone == "EST") {value = "(GMT-05:00) Eastern Time (US & Canada)"}
  else if (timezone == "CDT" || timezone == "CST") {value = "(GMT-06:00) Central Time (US & Canada)"}
  else if (timezone == "MDT" || timezone == "MST") {value = "(GMT-07:00) Mountain Time (US & Canada)"}
  else if (timezone == "PDT" || timezone == "PST") {value = "(GMT-08:00) Pacific Time (US & Canada)"}
  else {value = "---"}
  $(".timezone-div select").val(value);
}); 
</script>

What if we need to identify timezones around the world?

I can add more else if statements, but what are the time zone and value formats within the getTimeZone function?

We list all timezones in this article: https://support.desk.com/customer/portal/articles/2148835?b_id=7112