Not trying to nitpick, but what do you mean by a pushpin? Maybe it's just me, but a pushpin is a spring-loaded tool used to push headless/almost headless pins into wood. Mine is 2 telescoping brass tubes with a magnetic "chuck" inside. I don't think you mean cutting that do you?
If you mean cutting 6 mm off the length of a "steel" headless/almost headless "nail" I would say provided it's not more that about 1 to 1.5 mm dia, and provided it's normal mild steel (not hardened like those pins used for hammering into walls to hang pictures, etc, which normally look a bit blue in colour), and provided your side cutting pliers are in good shape then they should cut off the length of such an "ordinary" pin fine.
Try it a bit gently and if you feel a great resistance as you increase pressure (and/or if the pin is blue) then don't use your side cutters. Ideal then would be a thin cutoff disk in something like a Dremel drill running at pretty high rpm. If you don't have such a tool then an ordinary drill (battery or mains) will be fine, but due to much lower rpm will take longer.
If the worst comes to the worst you could use a hack saw (32 TPI blade) but even that will struggle on really hard (blue pins), and it you've got a lot to do, hack sawing will get pretty boring pretty fast!
As said above, if using side cutters then expect the bit you cut of to ping away somewhere where you can't fund it (or straight into your eye - so safety glasses).
You CAN buy cutters (plier-type) with specially hardened jaws but they're expensive and even with those you will struggle if your pins are hardened (blue).