There is easier way to use same Contact Form at Front Page and other pages. There is an option to put custom Form using third party Contact Form plugin. For example you install Contact Form 7 plugins, create and configure it. Then take its shortcode and out it at Appearance>Customize>Contact us section >Contact form shortcode(replace built-in) box. Then you will be able to use the same shortcode at other pages, posts, etc. The option to put custom form (shortcode)is available at Emmet theme Pro version.