I use flash quite often in the early morning, or late in the evening, or on overcast days. I use a Nikon SB 25 and a Metz 45 CT4.
A very important thing to remember is never use direct-flash. It just looks too obvious. Even if the main subject looks OK their eyes will usually glow or the background won't look right. I will however, use a bounce card to diffuse the flash to make it more natural looking.
I just started using this technique recently and it has made the world of difference. I experimented quite a bit last fall in Jasper and Banff photographing elk. On overcast days, or in the early morning, the photos without flash just never look sharp, or have any detail, due to the lack of contrast. The photos from the same series with the bounced - flash, were fine. The contrast was good with lots of detail in the animal. Most importantly, you couldn't tell they had been flashed.
Now keeping in mind that when you bounce the flash into a bounce card, you really cut down the working distance of the flash. That's why both flashes that I use are pretty powerful. I use the more powerful Metz for big game, and the Nikon for flowers and small game such as snakes.
Another important trick to using your flash outside is not to always use the highest shutter-speed allowable on you camera.
Most, newer cameras have very high flash-sync speeds 1/125th, 1/250th/sec., or higher. The problem with these high shutter-speeds, especially in the early morning, or at dusk, is that it usually underexposes the background to the point that the flash is now obvious.
I will often slow my shutter-speed down to 1/60th, or even 1/30th/sec., to allow for the background exposure.
Of course this technique only works on fairly stationary subjects. There are still times that you must use the higher speeds, but try some experiments.
I still prefer available natural light, but if it's a choice of bad available light photos or good flash photos, I'll definitely use the flash.
A cheap flash with a maximum working distance of twenty feet won't do you much good because by the time you bounce it, the working distance may only be about 10 feet. This is definitely not recommended for big game.