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Dunno about neatly. We just bodged a bit of sink waste pipe in which dumps the water into a corner of a raised bed. Not elegant but stops the shed sitting on a swamp.
If you want it to be tidy you could run pipe down the back of the butt to ground level, then across to the bed I guess. The pipe doesn't need to be downhill all the way as long as the exit hole is lower than the entrance hole.
If you use a diverter on the down pipe then you can achieve the same result by rerouting the remaining down pipe. If you do that, it's easier to remove the water butt in future as it's not really an integral part of the pipework.
That's a really good idea.
The gutter on my shed is currently directly flowing into a butt as a temp bodge until I redo the down pipe gutting and add the proper diverter. The veg patch is next to it and previously the gutter pointed a ΒΌ of the way in.
You've got me thinking that a perforated overflow pipe could distribute the water more evenly down the patch.
I guess it's just how to do it neatly.