In Dragon Ball Super Goku taps into an ability called Ultra Instinct. It's a fictionalized version of a real life martial arts phenomenon wherein you move via instinct without thinking. It's perhaps more commonly known as muscle memory. In DBS it allows him to dodge essentially anything and turn his dodges and counters into attacks with blistering speed and strength because he is no longer thinking. Goku is just...doing....he is riding on instinct alone.
When you take his years of experience against a vast array of foes and couple it with his speed, strength (which he usually subconsciously restrains) and most of all his spider sense (originally called his spider instincts in fact) Spider-Man could probably manifest something similar to Goku (obviously minus the stuff with ki).
The spider sense is itself directional and even works in his sleep so this is something that helps him auto-dodge stuff anyway.
He's got insane speed feats that really only active speedsters like Speed Demon can counter (e.g. he has out maneuvered guys like Fire Lord).
And he's way stronger than people think. In Spec #290 he was angry enough to just pummel Rhino into submission and he has tussled with the Hulk. His lifting power is even more impressive as he once helped support the Daily Bugle. It's also essentially agreed upon that he ALWAYS holds back his full power, e.g. in ASM #700 Doc Ock in his body casually punched off Scorpion's jaw by accident and Scorpion is supposed to be stronger and more durable than Spidey. In an issue of Venom by Daniel Way, Spidey was able to even over power Venom.
In the JMS run more than once Spidey sort of lost himself in battle and this was referred to as 'the spider' part of him taking over, which sounds a lot like Spidey was giving into base animal instincts to fight and survive.
For these reasons I think Spidey could when pushed fight in a similar way to how Ultra Instinct works and become yet more formidable.