I can relate to the fast time pressure and just keeping the first thing you made having a positive impact when making music. If you hear your own idea too many times it's very easy to get stuck and not know where to go from there.
The most typical and easiest would be a "Synth Pad" sound, these usually play long chords, only changing when the chords in the composition change. Usually that's when the bass changes but that's not a rule.
The more advanced option is a "Poly Synth" sound where you play more of a rhythmic idea or even a riff. The most basic form of this would be a "Stab" sound playing each time the snare drum hits. (and maybe with a few filler rhythms for variation.)
There's a bunch of different "Layers" that can be added depending on the specific genre of music.
Of course all of these should be layered and left out for contrast when the music needs it!
Edit: One more way to add harmonic information is an "Arpeggio" These are common in chiptunes/video game music since the old hardware couldn't play chords. Arpeggios add a lot of energy and forwards momentum to a song, maybe try them in some song too!