What's the best way to fit many components onto a Eeschema schematic?
I don't know Eeschema, but I don't think the actual EDA software is relevant (especially since Eeschema supports multi-sheet schematics).  

Don't try to cram too much on your page; give it some air. Net lines are much easier to follow if for every few lines there's some space between them. If your schematic becomes crowded you probably can divide it into functional blocks. Move one or more of these to another page, and use off-page connectors to connect the nets.  

The power supply is a good candidate for this, since you'll be using connectors to draw power supply connections anyway (and not line connections).

edit
And don't clutter your schematic with unnecessary information.  

  

This is part of an A4 size schematic which I had to blow up to over A2 size to be able to read it. (That schematic fails the olin test with honors.) Only show refdes and value on your schematic. The resonator bottom left for instance doesn't even show a refdes(!), but it does show pin designators and the product's ordering code. Don't! Just display "X1" and "16MHz". The pin designators "XTAL1" and "XTAL2" are useless, and the ordering code belongs in the BOM, just like things like package and such. Don't say on your schematic that a resistor is an 0603; you don't have to know that to read the schematic. Is useful for the assembly shop, but they'll only get the board layout and the BOM, not the schematic.
If the "CD1206-S01575" and "TS42031-160R-TR-7260" weren't there I might have been able to read it at A4 size.