Unfortunately, this is the cost/quality dichotomy we all face, particularly when starting out (as I am also).
It just can't be expected of a piece of kit costing less than a grand to perform the same as kit that will cost five times that. I'm sure you're aware of this, but I'm just stating the point.
All the same, hiss isn't going to ruin the recording (as long as it's not for commercial use). It's better to have the sound recorded than not!
You can try some low-pass filtering, or high shelf EQ, to minimise the noise, but it will affect the timbre of the birdsong.
Noise removal plug-ins will be fairly ineffective as any attenuation of ambiance that quiet will effectively kill the signal. (Someone may correct me on this)
Aside from this, get a piggybank and start saving for some Schoeps.