You could just add the input directly into your form, after copying the output from the form on the page.
For example... If you add the form tag
<label>The Date</label>
[text* the-date]
on the output of the form when you view the source on the page you've pasted the contact form shortcode, you'll see this:
<label>Enter Date</label>
<span class="wpcf7-form-control-wrap the-date"><input type="text" name="the-date" value="" size="40" class="wpcf7-form-control wpcf7-text wpcf7-validates-as-required" aria-required="true" aria-invalid="false"></span>
Then copy it back to your form and replace your form tag [the-date]
with:
<label>Enter Date</label>
<span class="wpcf7-form-control-wrap the-date"><input type="text" name="the-date" value="" size="40" class="wpcf7-form-control wpcf7-text wpcf7-validates-as-required" aria-required="true" aria-invalid="false" data-timepicker="true" data-language="en"></span>
Just make sure you include in your email [the-date]
- as it won't show up on the mail tab to remind you.