No, it is not possible to add a new value when inserting a new case. (Among other things, modifying the contents requires administrative privileges, which normal users don't have, but I suspect there are other technical reasons which make this difficult.)
It is possible to set this field on several cases at the same time through the FogBugz interface (providing that the field initially has the same value in all the cases, and that none of the cases have been Fixed). You could also do this with an SQL statement (after making a backup!) along the lines of
where, of, the table name will depend on when you deployed the Custom Fields plugin relative to other plugins, and the custom_field_name will depend on the field names in plugin_#_custombugdata.
It is preferable to use FogBugz to do this rather than to manipulate the database directly.