I've done this Gif from the Alyea film showing the moment Day picks up the rifle. Two things need to be noted:
1) When it cuts to Day he is already crouching down with his hands on or near the rifle. There is no telling how long he was down there
before it cuts to him.
2) At the very beginning of the clip the rifle is out of sight so we can't tell what's happening down there.
Conclusion - the Alyea clip cannot be used to support the idea the rifle was lying on its side. As soon as Day grabs the strap the rifle flips into a "lying down" position (I'm sure there's a better way to phrase it)
The rifle was already laying down when Day grabbed the strap. He actually grabbed the strap after he rotated the rifle towards him with his right hand (probably by the scope) before grabbing the strap with his left hand. He was pretending to be mindful of prints.
Had he grabbed the strap and flipped the rifle on its side then the stock would not rotate toward him as he lifted it up. The bottom of the rifle would come up 1st and the rifle would be upside down from the get go. Besides, I doubt Day would pick the rifle up by the strap and flip the rifle on its side. He may have laid the rifle on its side before picking it up, for some inexplicable reason, but his general handling of the evidence is damning. Everyone there was handling the evidence bare handed and made some token gestures about "protecting" potential prints.
Here is a shorter version of your GIF showing just the pick up. Day definitely grabbed the rifle somewhere with his bare right hand, rotated it towards him, then slid his hand under the rifle and grabbed the strap as he picked it up. Otherwise, the rifle would not have rotated towards him if he had picked it up by the strap.
MC pick upBut the elephant in the room is where is the strap in the still photos? Did Oswald take the time to hide the strap from view behind the rifle during his 90 second jaunt to the lunchroom?