It certainly gives that impression. In fact one might wonder why he didn't. Excepting that it was the middle of the night, and he was going to be busy the next day.
The noise? and it was in the wooded area, so could be difficult to get a digger in, without cutting down a lot of brush etc. Just a thought. And as you said The pheasant pen was fenced and it would need to be ...lots of foxes
Also
It would have taken him much longer to dig a grave by hand, so the body was laying about for some time.
He couldn't have killed her, had a drink, checked to make sure she was dead, decide what to do next, find something to wrap her in, wrap her in curtain, check the coast was clear, lifted her into the boot, gone to the pub, had at least one drink while deciding what to do. drive back and pick up a shovel, find a spot, dig a grave lift her into it and fill it in....in less than an hour, surely?.
He fractured her neck in several places, so she may have had blood coming from her mouth.