the rhino suit does not come directly after the armored car scene. no way.
spider-man traps aleksei on the street with his webs and walks away with the vial. the scene ends there. it makes little to no sense for aleksei to then break out, and immediately get (no scenes inbetween) a mech suit. it's ridiculous, especially when you have already had a whole bunch of action (chase, armored car shootout, truck chase, bus rescue, aleksei vs. spider-man outside the truck).
he's holding the same vial in this shot. further proof that the scene ends with spider-man pantsing aleksei. he probably runs out of web fluid and hitches a ride on the cop car, and calls aunt may or gwen or someone else, explaining he's running late.
rhino rampage comes later, after a handful of non-action scenes that set up the plot and characters. aleksei probably wants revenge on spider-man, so causes a ruckus that draws his attention. max may be met here, or earlier during the armored car chase.
edit: and spider-man definitely has no vial when he saves max