I used my alcohol stove all through the summer and had little trouble with it. It's a v8 can with holes punched around it just like this:
http://zenstoves.net/ChimneyStove.htm
But I set my pot directly on it without a stand.

I went backpacking a couple of weeks ago and in the cooler fall weather, I kept putting it with my pot. I finally had to let it burn for a long time before I could put the stove on top.

What design changes do I need to fix this problem? More holes? A taller can with the holes higher up? Something else?

Alcohol stoves do get more temperamental at lower temps, and they take longer to warm up. Keep the fuel and stove warm if it's cold (under your coat). I used an alcohol stove down to 2 deg C, but it took a long time to get it going and it never reached max performance.
Tom

It goes out because your cold pot sucks the heat away from the stove. With a thick gauge wire, make a V and put it on top of the stove before setting your pot on it. This will eliminate some thermal contact. You could use a wire hanger or a bike spoke or something. If that doesn't work, you should either make a pot stand for your stove or use a stove that requires a pot stand

What about a priming pan, would that help?

As others have observed there are three strategies to extend the conditions that alcohol stoves can be used:

1) keep the fuel in your pocket so it stays warm
2) insure good thermal feedback so the fuel doesn't cool off. Systems like the caldera stove help with this. Setting the stove on something that insulates it from the ground can be helpful in many situations.
3) A small priming pan to heat up the stove.

--mark

I'm not sure what a priming pan is. Is there a picture somewhere?

The think I liked about my stove was it didn't need a stand and everything I needed to cook fit inside my pot.

All the previous methods listed before should work but require you to add something else to your kit. I have found that when its too cold to set the pot on the stove right away, holding the pot ~1" above the stove while it's lit and pumping the pot up and down a little, helps to warm the pot, speed the pressurization of the stove so the jets flame faster and also helps warm your hands a little. I use a heineken keg can and a alchy stove made from the directions on thru-hikers site.

http://antigravitygear.com/proddetail.php?prod=TDPRPN&cat=102

The other thing to try is using a square of foil as a base for the stove to keep you from losing the heat into the ground, if you don't already. If you use heavy foil you can use it as a priming pan too.

Thanks for the picture. I had an Etowah stove before I made my own. I guess I can use its outer part as a priming pan or else use the whole stove again. Thank you!

As others have mentioned, insulation between the stove and ground is key. The stove takes some time to warm up enough in colder temperatures. A wider pot helps to absorb more heat, as does have a good, tight foil wrap around the stove and pot base to maximize the limited btu output.

The coldest temp I have used a Trianga in was 17 degrees. It took nearly 18 minutes to boil the water.