If the 4400 isn't clipping at high signal levels, you probably don't need an add-on amp. However...
For movies, I'd suggest a 3 channel amp, as the center does a lot of the work. Unfortunately, 3 channel amps tend to be more expensive than stereo (2 channel) ones.
Class D amps use switching mode. (Sometimes erroneously referred to as digital. They are not; analog input, analog output. https://en.wikipedia.org/wiki/Class-D_amplifier
Some audiophiles wouldn't touch them, but they are the most efficient amps available.
Crown makes "pro" audio gear. That's not primarily intended for home audio, but people use it for that.
I'm using an Emotiva XPA3 Gen 3 with an AVR-X4300H (immediate ancestor of the 4400). I'm happy with it, but it's expensive. It's basically a class A/B amplifier, but it uses an active PSU to improve its efficiency significantly.
A Monoprice Monolith 3X200 amp is slightly cheaper, but not much.
A Crown XLS series amp (https://www.crownaudio.com/en/produc...ecore-2-series
) would be easier to use than some "pro" amps: it has RCA inputs and binding post outputs, in addition to its XLR inputs and Speakon outputs. I haven't used one. They are only stereo, so you'd need two (and not use 1 amp channel), Or, you could buy 3 and use them as monoblocs in bridged mode.
Later: one drawback to the Crowns: no 12V trigger, to allow them to be controlled on/off by the AVR. There may be a way to achieve that using the "aux" port.