It looks like if you add a custom field in JIRA, Aha will not recognize it unless you delete the integration and add a new one, which means you need to relink everything.
You should be able to go through the integration UI and add new field mappings including the new custom Jira one without loosing all existing links.
The "Reload configuration" link at the bottom of the mappings page will fetch the latest configuration information from Jira, allowing Aha! to know about newly added custom fields.