First (bottom) layer has gaps First off - I'm rather new to this and I might be on the wrong track altogether...

I'm printing PLA using a simple DIY XYZ printer with a direct-driven extruder, 0.3 mm nozzle, 0.2 mm layer heights, using Slic3r. I've tried to calibrate the bed as level as I can. I believe the distance to be OK because the skirt prints in straight lines with no wobble.

While the overall results aren't that bad, I'm trying to improve the quality. I'm not happy with the first layer: While it sticks to the bed nicely, the printed strands are too far apart - there's a very noticeable gap between them that I can actually see the second layer through. There appears to be a problem with the second layer as well which always ends up too small (recessed). That's only the second layer, though - from the third layer on, that problem disappears. 

What I've tried so far:


change the nozzle and/or bed temperature by 10-20 degrees up or down - no visible change
change the print speed (using the dial on the printer) - no noticeable change, especially when going slower
change the first layer extrusion width from 200% to 250% or 300% - that made the strands "flatter", but also made Slic3r space out the strands so that I ended up with gaps again
increase the extrusion multiplier - that seemed to help the first layer, but lead to over extrusion further up, so I didn't pursue that any further
manually increased the flow setting for the first layer in the printer menu to 135 - that seems to do the trick, but I have to turn the setting back again on the second layer


Now - is this the right way to fix this or am I just patching over an entirely different problem that I just failed to diagnose properly? If it is the right way, how can I tell Slic3r to either change the flow rate / extrusion multiplier just for the first layer or increase the extrusion width without spacing the strands further apart?
You can, in most slicers, set a separate extrusion multiplier for the first layer. This doesn't appear to be possible in slic3r, but there are a couple of ways to work around this:


Change the Z offset (either in software or by adjusting the endstop). Bring the nozzle closer to the bed for the first layer. You can combine this with an increased first layer height. This is the most appropriate fix for your issue.
Use M92 EX.XXX to increase the extruder steps/mm in the start G-code, then set it back to normal in the layer change code. This emulates increasing the extrusion multiplier for the first layer.