How do I create a new custom field or update existing field values in a company-managed space?

Question: Is it possible as a project admin to create a new field only for my company-managed project? If so, could you please direct me to where I would do that?

Context:

I have project admin access to the space, but not global. We have a specific field for labels I'd like to update the field values for.

Looking into this, it seems like I do not have the ability to do so directly within the Jira UI (~ a "manage fields" page), since this field is global. As a result, I'm looking into another option of creating a new field entirely to eventually replace the original field.

I tried adding a new field from the request type, but also do not see the option to create a field here - just the message "Can't find a field? Contact your Jira admin"