Yes, but it depends on the provider and where they place the CID and what is being used to send the forwarded call (SIP,Analog, PRI, etc). There is not a single universal setting that can be employed for all.
You can look at the trunk settings, outbound parameters and then the SIP headers labeled starting with FROM. Make certain that you have a current backup or at least copy the parameters as they exist before making changes.
Using the pull down there will be a selection that has "Originator caller ID........". You can either use wireshark and see which field is being user of the 3 labeled with From or you can simply start at the top and try them one at a time.