It's gotta have larger iron.

A more robust power transformer to supply the necessary extra current, and to run at a higher plate voltage to so that these 6L6GC's will actually provide more output.  The GZ34 rectifier will help get that plate voltage up even higher.  Without crunching the math, I'm guess plate voltage may be in the 425VDC range (5E3 is around 370VDC), the switch to the GZ34 alone gains 33VDC. And then a beefier output transformer that can take advantage of that extra power. 

Popping 6L6GC's in a regular 5E3 circuit doesn't get you much, only a few extra watts because the small, stock 5E3 output transformer chokes it off and the plate voltages aren't taking advantage of the 6L6GC's capabilities.  But the 5X3 should put out around 25-30 watts RMS depending upon voltage (5E3 is 12 watts RMS).  It'll also be tighter, with less sag, and the 6L6GC will provide bigger, more focused bottom-end and more top end.  Perhaps, one could drop a KT66 in there and get to output will retaining the 6V6GT midrange (6L6GC's have a scooped midrange designed into them).