If you want to be tracking a value over time using the timestamp option then no you currently can’t force it to animate on page load as it will only animate every “tick” which may not be noticeable depending on the interval value and tick_multiplier.
However, if you just want to use it for visual effect and you are happy to update the end value manually if it ever needs changing, then you can do something like this:
[jellyfish_counter start=533290 end=533301 tenths=false digits=6 speed=100]
which would show a simple counter that animated from 533290 to 533301 every time the page loaded. See an example here https://demo.strawberryjellyfish.com/2016/04/simple-count-effect/
I suppose it might be an interesting future feature to add to the plugin that started a timestamp based counter slightly below the actual current value and animated to the actual value on page load, which I think is what you are maybe looking for?