TT, we've had that problem in the past. Could you get around it by ordering a skip, or one of those garden waste places that leave a hessian bag on a frame and only charge you to collect when it's full? Sometimes those Jim's Mowing type places will do rubbish removal for a fee as well. Otherwise, is there an alternative recycling facility that would buy paper by the kilo? Scouts/schools collecting paper for $$? Other things we've done - chat to neighbours if you have them - about back-filling their bins after they've put them out on collection day (assuming you have rubbish collection). We've also bagged rubbish (pay more and get the heavy-duty garbags with the ties, etc), or boxed it up, and them "queued" it up in the carport/garage/along the side of the house. That way, you can dump the first lot as soon as the dump opens, then scoot home and collect the rest. Obviously this only works if your rubbish is "clean" of regular household waste (dont' want it being stinky or attracting critters) and if you make it a priority to get to the dump as soon as you can.

ETA - boxes are best if you have a lot of paper as it can be very heavy.