Iga

quincy is showing you why the studio folks earlier were suggesting to improve the lighting so that you can use more dof (higher f stop) to get more of the bird in focus without the image turning dark or needing to push the ISO. If you do go the lighting route, you have more options in lens choices -- though that 90mm is pretty good, a lot of portrait photographers like it. But one thought about that, try to get constantly on lighting instead of flashes. If you start strobing the units to get several good shots, the birds may panic, that sudden burst of intense light is startling.

According to Fujifilm, the XF 18-55 at 55 mm has a maximum magnification of 1:6.67, while the XF 90 has a maximum magnification of 1:5. This means, when both lenses are at their close focus distance, the XF 90 should see a slightly tighter crop. The image area in the plane of focus of the XF 18-55 at 55 mm should be around 16.7 cm x 11.1 cm, while the image area of the XF 90 should be ~ 12.5 cm x 8.4 cm. I've added those frame lines to your image:

 
But please also consider that your depth of field gets smaller as your magnification increases. So even if you buy a lens with better magnification and brighter aperture, you might still end up closing the aperture down to get the whole head in focus. I've added some pictures I took with the XF 80 f/2.8 wide open at f/2.8: (the last bird might even be a pigeon, but you'll know better than me)




You can see that at f/2.8, the feathers at the back of the heads as well as the beaks are already very unsharp and out of focus in the portrait images while the eye is sharp. Even with less magnification as seen in the last picture, you can run into those problems.

You might be frustrated with hunting autofocus of 60mm macro while framing moving subjects though. It's good with stamps or coins, not sure about pigeons.

Can you rent a Fujifilm 90mm F2 lens from your local camera store? It is bright, has good background blur, etc. etc. and allows you to focus from 0.6m - ∞.
Samyang / Rokinon also has a manual focus 85mm f1.4 that gets good reviews and is less expensive to buy.
Before you go the new macro lens route, you might also try extension tubes for your current lens.

The Fuji 90 mm f/2 has approximately 12.5 cm x 8.4 cm field of view (1:5 magnification) at the close focusing distance of 0.6 m. I'm not quite sure if that's enough for "head and shoulder" portraits of birds, but it might work.
The 80 mm f/2.8 macro will do 2.51 cm × 1.67 cm (1:1 magnification), which is definitely enough, while the 60 mm f/2.4 macro can reach 5 cm x 3,3 cm (1:2 magnification).
(The Zeiss 50 mm f/2.8 macro can also reach 2.51 cm x 1.67 cm, by the way.)
Your 18-55 has a maximum magnification of 1:6.6, so if that was enough for you, the 90/2 should work.
 
Right now, I don't think there is a native Fujinon lens with an aperture of f/2 or brighter that can focus closer than the 90/2 can. (Except for the 16/1.4 which will distort your birds badly).
 
I think any of the three (60, 80 or 90) could suit your needs, or you could really just add a flash or continuous light with softbox and stay with the 18-55.
 
If manual focus is an option, there are lots of third party lenses you could adapt or even get in FX mount.