Use the EQ on the input-side of the recording chain,...
to tweak the sound to be as optimal as possible, before recording to tape.
It doesn't hurt to give the treble (Hi-EQ) a little extra bump up, for pre-empasis of the high end, 'cause there's a natural rolloff of the high frequencies when recording to cassette tape. That's a LITTLE bump in treble for pre-emphasis, & nothing too searing!
Also cover the basics,... that your heads are thoroughly clean, that you're using high quality name brand Type II tapes, that the dbx NR is switched ON during recording and playback, and that you're recording on the High Speed setting.
I also agree that the 424mkIII is a fine machine that gives good sound quality, but a lot of the sound quality you get is based in technique, not technology.
/DA