I see movies with operators running with two jerry cans...um, yeah, right.

Ahhhhh ha ha ha. Yup. I've seen those movies too and those scenes were exactly what I was thinking about when I wrote that comment.

It's good to mention such reality for anyone else reading this thread. :-)

but I'm pretty sure I'm going to have to empty it, clean out the container, and start again with rainwater.

Before you do that, perhaps it would be a good experiment to take a sample of that water to see how much algae is in it, chlorinate it and wait a few days and then take another sample to compare...

...the unspoken truth about water treatment is all the "stuff" you kill generally only adds nutrition and protein to the water supply. It's best not to dwell on it, but it's the truth.

but rather make some kind of cone

I bought six of these massive latex basins for rainwater collection. They are huge, rubbery, two foot diameter, slot inside each other and cost $5.50 USD each.

I'll be experimenting with them during the next rainfalls to see if they offer an easier solution than butchering one of my down-pipes, but mathematics suggests they will be a fantastic and easy solution.

I'm currently keeping them on the top of my prepper pantry with electrical equipment inside: they effectively take up no space that way when you buy an additional basin and store stuff in it.

Perhaps a similar solution would be better for you than a cone? Just put them outside when rain is forecast and fill your water butt then put 'em back into storage.