Your overhead pressure will likely be determined by the nearest bottleneck in your overhead vacuum system, or some combination of any side cooling you have. Typically the bottleneck is some combination of the cooling media temperature in the summer time, and the approach temperature of the lowest pressure condenser, but it could be the vacuum system pumping capacity itself. However, in order to get there your top most side cooler and probably any lower side coolers must increase their heat removal. Otherwise if the side cooling is not increased as your column pressure reduces heavier and heavier molecules will vaporize and boil off over head, adding their mass flow to the vacuum device and adding more condensing duty from the current overhead cooling needs. It is like either a condenser or vacuum device is already rate limiting your column performance, so you need to find a way to reduce that incoming vapor load so the pressure can go down further
If you increase the side cooling, any side draw products you are currently producing will become lighter in average composition. Especially the front end of their boiling curve.
Get a precision distillation of the feed stock to this column paying special attention as your labs is capable of doing to define the front end boiling point, the compounds in that initial boiling mass. Get a component definition of the first 15% if you can. Collect flow an composition data on the vacuum system exhaust gaseous and liquids leaving the condensers and compare that to what your lab results are.
Then simulate the full distillation curve to your corrent conditions, adjusting the condenser duty, and pressure to replicate what you see happening today. Note the vacuum device mass flow momentum and the condenser duites. Then figure out where the cooling has to be increased while those bottlenecks are held constant or reduced. Look for ways to increase side stream heat removal, this accomplishes two things it pulls light ends out before they tax the overhead vacuum system, reduces the overhead flow and condensing duty as well.
A word of warning, it is possible to get low enough in pressure that the system could start vaporizing compounds that must then condense and some will even solidify in the condensers. Paraffins are a particular problem for vacuum condensers. Once paraffin or high viscosity compounds coat the condenser cooling surface your cooling capacity is greatly reduced and the vacuum system could start surging due to the hotter, then colder cycles as the paraffin is melted off, then replaced anew.