How can I add/rearrange custom fields in the default contact view template?
The template ​templates/​CRM/​Contact/​Page/​Inline/ContactName.tpl does not contain any custom fields. Instead of adding smartyDebug=1 in the url, which dumps variables in the main template, try adding the smarty tag {debug} in this specific template to see the available local variables.

At minimum you will probably find the contact id, and from there you could use the api to fetch necessary data (note the api explorer can generate this syntax for you):

{crmAPI var='result' entity='Contact' action='getvalue' return="custom_1" id=123}