No, leave those 1uF caps - they still serve the purpose of draining RF noise to ground.
Now that I think of it more, I'd ground one side of the filament winding. Any chassis ground will do, it does not have to be signal ground. Even better, use a pair of resistors, for example 100 ohms, one from each side of the 6.3v winging to ground - this makes a sort of center-tapped winding which will probably reduce the hum a bit.
If there is audible hum, that center tap can be connected to a voltage divider on the power supply, to bias the heater power to around +35 volts, which would probably make another improvement. The 35v tap on the divider would have to be bypassed to ground with a capacitor, perhaps 100uF electrolytic. But I'd recommend that you try it the easy way first and see if that's good enough.