Slash chords are often denoting the inversion to be played, as in Am/E, where on guitar, for example, the player would use the open E as well, it being the slash part of the chord. Othe slash chords might use a walking bass line approach, as in C, C/B, C/Bb for an effect to get to the IV chord of F.So, the slash part actually defines a chord more learly - C/E will be 1st inversion of C; C/B the 4th inversion of Cmaj7. Occasionally the 'slash note' will be foreign to the chord, but often as not, it'll be part of a bass line.
Sounds like you need to learn some more chords! Dims and augs are often the 'missing chord'!
As a basic starting point, you need to be aware, and able to play, all inversions of major and minor triads- as in C-E-G, E-G-C, G-C-E. There's no clue in the question as to whether it's piano or guitar you prefer, but guitar will inevitably cloud the issue, as the inversions are not as clear cut as they are on piano.
Assuming piano, then get your head round the 4 (in)versions of the dom 7, maj 7 and min 7 chords, only using one of each of the 4 notes involved. Be aware that the VOICINGS can be open or closed, which again muddies the water... Dims and augs need sorting too.
As a basic premise, look to the 'main' notes in a bar - often on 1st and 3rd beats. These will give a good idea which chord may fit that bar. Generally, the chord will contain those notes - otherwise the melody or the chord is wrong! Also be aware of the progression - the chords either side of the elusive one. Look to the oft-used pattern of ii-V-I for a route through.
As a little example, let's say there's a C and an E in a bar.(No, it's not the beginning of a joke...) Chords which may fit are : Cmaj.,Cmaj7, Amin.,Am7, D9, Dm9,Fmaj7, Fm maj7, Ab+,F#m7b5, to name but a few. Faced with C and E in a bar, then, depending on key, and preceding and following bars, one or more of these may well do the job!
Bottom line is - try loads, and the ones that sound good will be good!