I've made a root ginger concentrate which came out pretty well (bashed up a quantity of fresh root, warmed in PG and steeped for a few days, then strained through a coffee filter). I expect you could do similar with the stems, although with green source material you would need to be careful not to over-steep it: green vegetable matter will develop an unpleasant cabbage taste if it's left steeping too long - which would be more than 3 or 4 days.
I was pretty happy with the strength of mine after the first time, but if you wanted it stronger, bash up a second batch of ginger and repeat the process using the previous batch in place of fresh PG. (Had to do that with some dried mint one time - it was fine after the second steeping, but there's a significant reduction of the final volume, due to the fraction of liquid that will not be persuaded out of the soggy mass of ginger pulp and the filter paper).