If you read a description of a character moving spider-like, how would you picture it? Running fast but with purpose or in a jerky, stop start motion?
Scuttling along on all fours, quickly and nimbly, darting from place to place.
Our house is host to numerous wild spiders. None of them ever move in a jerky way. They're always very adept at getting where they're going. But they do stop and start a lot as they dart from spot to spot. It's not jerky or halting, it's just...well, stop-start-stop-start-stop-start, dart dart dart. Zip zip zip. Almost like they're not moving at all, but are teleporting from one spot to another. When they're moving a longer distance, they scuttle, legs all trundling along rapidly, but still gracefully. Ziiiiiiiiiip.
And in truth, I would never describe somebody running on two feet as moving in a
spiderlike fashion. They'd have to be on all fours. But that's just me.
ETA: Hey cool! I didn't even read the previous posts and I see "scuttling" is already repeatedly mentioned! I have to add, too, that the strides (while crawling) would be very wide--very long reaches with arms and legs.