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If I can understand, you want to send a form to the contact info you collect from another form, I think one of the data you will collect will be the email address, then if so , you can share the link to the form and the user will be able to fill it out, to do it, please follow this guide :
Hope I understood your question and if it doesn't help you, please feel free to contact us and share us more details about what are you trying to accomplish, we will be more than happy to assist you.
It seems that your last reply did not came trough, so I will quote your reply that I found when I checked our email log.
"Is there a way to do this automatically or would I need to send the link over to each person that submits that contact form?"
You can set Autoresponder email inside of your form and add link of your second form inside of Autoresponder. That way when your users submit your form they will receive data they have submitted on your first form and link of second form. Inside of this guide you can find how to create a Form Autoresponder: http://www.jotform.com/help/26-Creating-a-Form-Autoresponder
You an also add link of your second form on thank you page of your first form or you can redirect users to your other form once they submit first form:
Hope this will help. Let us know if you need further assistance.