For most Conga Composer solutions, it’s often desirable to pre-select the template that you’ve intended to use with this particular merge operation. You may do so by including the TemplateId parameter in the URL that calls Conga Composer.

https://composer.congamerge.com/?SessionID={!API.Session_ID}
&ServerURL={!API.Partner_Server_URL_290}
&Id={!Contact.Id}
&TemplateId=01H500000005tMS

The ID of the desired template is found by navigating to the template, stored in either the Salesforce template repository or the Conga Template Manager.

Click the name of the template and select the ID from the template’s URL.

The ID required is from the Template Manager record, not the attachment stored with the record.