Possibility to set Zendesk Talk number as caller id when using overflow

We are using a 3rd party service to pick up all our support calls. We then using the Overflow feature. The issue is that the #rd party service use the same number for many different customers, and they need to know that the incoming call on their end is forwarded from us. As described in your documentation the standard feature is “caller ID sent to the overflow number will be that of the original caller”. I would like an option to choose wether the caller ID sent to the overflow number should be the original caller, OR the Zendesk Talk number.