Green’s Dictionary of Slang

(Aus.) a beer glass that is small than a schooner but larger than a middy.

2001		Google Groups: alt.homebrewing 9 Apr. 🌐 [S]ome yuppy pubs sell beer in a glass that’s a cross between a schooner and a middy, called a schmiddy (I shit you not). I don’t know the exact volume. I won’t drink in pubs that serve schmiddies.
2003		Aus. Word map 🌐 schmidee: Size of glass used to serve (normally) beer. A little smaller than a schooner, a bit bigger than a middie, hence a schmidee.
2006		Sydney Morn. Herald 6 Nov. 🌐 [blogspot] [N]ext thing you know we’ll be inundated and the publican will think about spending money on refurbishments and chrome chairs and tables and extra lighting and a bouncer and increase the price of beer and replace the schooner with expensive schmiddies and start stocking trendy boutique beers and mixers and attract the type of clientele that drink boutique beers and mixers!
2017		Aussie Home Brewer 25 Aug. 🌐 Then there’s the ‘schmiddy’ where pubs serve you something a wee bit bigger than a middy (pot/half/handle/SA schooner) and charge the same price as a schooner (SA pint).