Some discounts do allow for a reduction in fare on a current reservation - particularly FL resident rates and MIL rates have historically allowed this, sometimes even KSF. In that case, then yes you can request a refund (or onboard credit) after PIF. If it's a special discount - like *GT - that requires a new booking, then you would have to cancel the current reservation and rebook. It's all in the language of the discount offer whether or not it can be applied to current bookings.
If you pay-in-full before the PIF date - you should be able to get any fare reduction that is available to be applied to a current reservation (FL, MIL) as long as it is still prior to the official PIF date. You should even be able to cancel and receive full refund before the PIF date.