I hear people whine about playing in different keys all the time (Such as a C#), and I don't get it. Either I'm doing something wrong or I just get this concept and know my notes better than they do. If I were to play in the key of A minor, I could just play any note in the A minor scale, right? And the same for any other key. The piano player in my jazz class makes such a big deal out switching keys (pun not intended), and I'm wondering if there's something I'm not getting.
Some songs change keys, I really see no need except when it accomplishes something. Think about this. The vocalist normally decides what key they want. Why in the World would they want the song to move to another key? Now Jazz will do this more than other styles, however, here you have more solo instruments than just a vocalist, aka horns, sax, etc. and the horns, sax, etc. have keys they like to use - mostly the flat keys. Then there is that mood change - major is happy and minor is sad, so there are reasons to change, but, IMO there should be a reason for the change.
Moving on to what the other posts said about there being more to this.....
I think you already understand this and was using a generalization in your post. If not read on.....
I'll give an example. The song is called out to be in A. {edit} Just saw your example is Am and I'm using A major, humor me.
Those playing melody can draw their melody notes from the A major scale as you said. However, your bass line should provide harmony and that is normally done with the chords made from the A major scale.
Which chords? The ones used in the chord progression - for this specific song. Lets assume that will be the classic ole I IV V7 progressive in the key of A.
I......IV.....V7
A.....D......E7
When the A chord is active in the song our bass line could contain the chord tones of the A chord, A, C#, E or we might play the A major pentatonic scale under the A chord, A, B, C#, E, F#. Or we could just play A notes, i.e. A, A, A, A, or the root and five, A-E-A-E, or the possibilities go on and on......... We want to harmonize what is happening in the melody. We do that by sharing like notes. If the melody line and the harmony line (bass line) share some like notes we harmonize (sound good together).
What I'm trying to point out is our bass line normally does not play random notes from the A major scale. We play specific notes from the A major scale in a groove that 1) harmonizes the melody, 2) maintains the beat and 3) calls attention to the chord changes.
www.studybass.com goes into detail.
Basic Chords
• Major Triad = R-3-5
• Minor Triad = R-b3-5
• Diminished Chord = R-b3-b5
7th Chords
• Maj7 = R-3-5-7
• Minor 7 = R-b3-5-b7
• Dominant 7 = R-3-5-b7
• ½ diminished = R-b3-b5-b7
• Full diminished = R-b3-b5-bb7
See a chord and play it's chord tones. As every key will have three major, three minor and one diminished chord it's a good idea to get your major, minor and diminished bass line chord tones into muscle memory so when you see a chord your fingers just know what will work. Now the song may only give you enough room for the root, or root five - adapt and get as many chord tones into your bass line as needed. Root on 1 and a steady groove from the other chord tones plus something to call attention to the chord change will keep you gigging.
Have fun.