Like jackal said, never appear that you NEED the vehicle, only that you are interested in driving something different. I once showed up at the Thrifty counter in YYZ with 5 pieces of luggage balanced precariously on my cart. The location had run out of Fullsize cars for the evening, but insisted that I wouldn't have been able to fit my things into a Fullsize and needed a bigger car anyways. Since it was nearly 1 AM and I was exhausted, I took the bait and got put into a Pacifica for $20/day more, even though I was actually alone and my luggage was only going to my mom's place 15 miles away.
That's why it's a good idea, dpending on airport configuration (I've never been through DEN), to leave other travellers to collect luggage while you fetch the rental. That way you could be anybody at the rental counter. If the agent sees you with 4 other people and a pile of bags, and you ask about SUVs, the rental agent is thinking "This guy is going to be CRAMPED in a Corolla/Caliber/Whatever. He needs a Grand Cherokee/4Runner/whatever. Let's see how much I can get for it."