Falcon03, if I understood your query correctly, you want to raise the BP of the side stream fraction, while keeping the degree of fractionation (ASTM gap). Is that so ?
An increased (normally 3-5% vol. on crude) overflash, i.e., heating coil outlet temperatures, could also be used to increase vapour loads.
Internal reflux is used for fractionation (ASTM gaps) by condensing upcoming vapours. This internal reflux is generated by the external pumparound, which is normally fixed by the heater's outlet temperature. Other heat is removed by the product streams. Temperature and (side-strippers) stripping steam rates are then the variables left to meet the heat balance requirements imposed by the overflash.
Besides, if the tower section washed by the pumparound is not trayed but packed, it is important to keep an efficient liquid flow rate for good spray distribution.
When you say "freezing" do you mean pour point ? If so, do you know for sure where the waxy paraffinic components are concentrated ?