Recently I transcribed this song here (with its original, the most common known chord progression). Now I will show you possible options of reharmonization – or, putting it simple – how you can change chords, so the song would sound (we also could say "flavor", "taste") a bit different.
I will try avoid much of the long theoretical explanation and show you just examples of different harmonic approach. So here will follow short descriptions; for examples themselves – scroll down. In a case you don't really care about theory – skip it, follow to the music.
1. The "original" or the most common harmony consists of simple triads – minor or major.
2. All the major triads here are substituted for dominant sevenths chords. They make the song sound a bit "sharper" – because of the tension to resolve in dominant seventh chord itself. E7 chord* is the most natural to substitute for, because it is a natural dominant chord in the key of A minor.
3. All the simple triads are substituted for triads with added 2nd step (or scale degree so to say) – except for the E7 chord . Each chord usually corresponds to a certain scale and is built from this scale's tones. For example – a C major chord belongs to a C major scale, which consists from tones C, D, E, F, G, A, B. The triad itself is built of C, E, G which are I, III and V steps relatively to the scale. Adding a 2nd step to a simple triad (the note next to a tonic, or chord root) will enrich these chords and will give them a certain flavor.
4. All the triads are substituted for suspended chords (sus). That means that the 3rd step in a chord, which makes the triad "major" or "minor", is "suspended" – substituted for a 2nd or a 4th step (and in case of dominant seventh' chords – for a 4th). These chords have a lot of tension and "strive" to resolve to their triad (or seventh chord) "originals".
5. A version which is similar to the original one – except for a bass movement in the first and the third lines (the bass line is the same, but chords are different – triads in the first line, dominant sevenths in the third). So we have triads or dominant sevenths again, but some of them are started not on their roots, and functioning more as "passing" chords. In the fourth line the Am triad is started with the 5th scale degree (E in this case), then goes to E7 and then back to tonic (Am).
6. A "jazzed up" version. There are no triads anymore, only seventh chords of all kind – minor, dominant, suspended and also with characteristical "jazz" extensions – 9ths or 11ths. Actually these extensions are simply 2nds (from the root) or 4ths (and in some cases 6ths), but they are called 9, 11 or 13 when the 7th is present. Too much explanation is needed here :) – just play the progression, check and listen how it "feels".
7. All the above stated examples are not written in stone. That's why we call it "music theory", not "music truth". Feel free to experiment, there are much more ways to reharmonize the song, I listed only the most obvious ones. Here's a combination of what you already saw. Remember – it is a matter of individual taste and knowledge :) Good luck.
* - if this tutorial is transposed, E7 will turn to a corresponding dominant chord (5 scale degree) of the appropriate key.