Not necessarily (well I guess it depends what you mean by "pushed"), but I think tube amps naturally boost the even-numbered harmonics at all volume levels. This sounds good but is distortion.
That’d show up in the Total Harmonic Distortion figure for the amp, and for any halfway hifi quality oriented amp it’ll be well below 0.1%. Perhaps “golden ears” can hear that? I’m not convinced. Most hifi audio amps typically run at well down into the single digit percentage of their rated output at ordinary listening levels in most systems, so they’re an order of magnitude or two away from clipping. If the feedback circuit and the output devices (transistors or tubes) can’t keep the response perfectly linear the thd down under 0.1% across the entire audio frequency band in that power range, it’s just badly designed.