Classic Unders by Hanna Andersson, http://www.hannaandersson.com/pdp.aspx?from=SC&pcid=28&styleid=37963&simg=37963_001&sz=XS.
Although one of my children was so tiny when she started that the Classic Unders were a tad too baggie on her. I ended up ordering some smaller ones from Next UK. The smallest size they had was 18 months but they fit my rather tiny 10 month old perfectly, http://us.nextdirect.com/en/x5d3628s2#850881. They’re 100% cotton and quite nice. Not as easy as the Hanna Unders for the child to get on and off, though.
For car travel or long trips to the grocery store I liked the Imse Vimse underpants the best, http://www.amazon.com/Imse-Vimse-Organic-Cotton-Training/dp/B0017RSXWC/ref=sr_1_9?ie=UTF8&qid=1431978674&sr=8-9&keywords=imse+vimse.