This theoretical discussion assumes we are talking about a well designed, good quality SMPS, not a cheap no-name one from Ebay!
If the Phase to Phase voltage is 440V RMS then the Phase to Neutral will be 253V RMS and this is what many domestic single phase supplies deliver, although nowadays it's specified as a nominal 230V. So a single phase and neutral is essentially what you have now as the input to the SMPS.
Consider a full wave bridge rectifier, this is almost certainly what your SMPS currently has at its input. This would deliver about 358V DC if supplied by 253V AC.
Now consider adding a second (and third) bridge rectifier to the other two phases and commoning their DC outputs. The DC output voltage will remain the same, but will contain a lot less ripple.
simulate this circuit – Schematic created using CircuitLab
So, at least in theory, you could substitute a 3-phase full wave bridge rectifier for the existing single phase one.
BUT it doesn't end there. The SMPS will have common mode filters and class-Y capacitors on the input before the rectifier and these should be duplicated on all 3 phases as well. Getting all these additional components safely into the original case would prove quite a challenge.
Smoothed rectified 3-phase is nominally the same as single phase (with less ripple), but would tend to be slightly higher as the extra phases will "fill in" the dips. Again a good quality SMPS would have components following the rectifier of adequate rating so they should (no guarantees) be able to cope with the potentially higher DC voltage, but you would need to check very carefully first.
All in all it doesn't seem worth the effort. I can't find any references to this particular circuit online, the nearest are either 3 phase half wave rectifiers or circuits involving transformers with 3 centre tapped secondaries with the centre taps commoned giving 6 phases which get half wave rectified. There appears almost always a transformer involved and nobody is advocating direct rectification of a raw 3-phase supply.
I am sure that there are plenty of power engineers out there who will tell you exactly why you shouldn't do this!