Important note: Conversions on the second Offer template will not be attributed to the Composer Experience through which you're triggering the initial Offer template.
To achieve this, you can add an external-event to the button in the Offer template. For example like this:
HTML
<input type="submit" external-event="open-template">
Then you will need to add a handler for this in your Composer integration script or anywhere in your website's JavaScript:
tp.push(["addHandler", "checkoutCustomEvent", function(event){\n switch(event.eventName) {\n case "open-template":\n tp.offer.show({offerId:'<ID>',templateId:'<ID>'});\n break;\n }\n} \n]);
You will just have to specify the values for the offerId and templateId. Please see our documentation here for more information.