Never even thought of this - will give it a try.
I was running 35psi previously (from advice I got on here) and when it got cold that pressure dropped down to around 31 psi. At around 31psi I noticed the ride got squishy and a bit washed out when turning. Going over bumps was nice and soft though.
I recently got my spare mounted at Discount Tire and they told me the correct pressure was closer to 45psi - inflated to that now and the ride is STIFF.
@bloc 's advise is good. Definitely worth experimenting and tweaking tire pressures. The RCTIP bamboozling on these boards has been a disservice as it misleads what tire pressure is and should be, because tire pressure was never borne singularly on load rating. Even OEMs tailor pressures below load rating at times to balance trades, and will recommend additional inflation pressure when laden. When really laden and towing, that is still good practice even if there is enough tire pressure to meet load requirements.
One caveat is that pressure should generally not be below the door jamb stated pressure (33 PSI) in regular driving use, for stability and handling, even if the tire carcass is good from a load rating perspective.