Help with sourdough bread
In my experience, the two things you can do to most improve the crumb of your bread are:


Autolyse - Basically this means that once you combine your
flour    and water, let the dough sit for awhile (I usually autolyse
for 20-30    minutes).  This allows the flour to more completely
absorb the water,    and results in a more extensible dough.  Some
say that you should    wait to add salt until after you autolyse,
but I have seen no    difference after trying it both ways.
Take your time - An extended period of raising and proofing your dough almost always results in a better and more flavorful loaf.  Don't hesitate to use your refrigerator during your first or second rise to delay the activity of the yeast.  For sourdough, I prefer to use the fridge for my final, shaped proof.  This has the additional benefit of some nice effects with the crust.


There are quite a few other factors that may be contributing to your difficulties.  It sounds like your starter may not be active enough when you mix everything together. I would recommend more time feeding and preparing your starter before using it.

You might also benefit from some additional tricks and techniques.  Using your list of ingredients, the method I would use to bake bread is as follows:


Combine your flours and water (Yes, you may be autolysing for several hours, this is a good thing).
Activate your sourdough starter.
Combine your starter, the autolysed dough, salt, oil, and honey.
Knead. (I use a stand mixer on medium and go until the dough pulls away from the sides. Be patient here and try to resist the temptation to add more flour at this step, unless you've kneaded for 10 minutes and seen no change, but then only add a tablespoon of so.)
Place your dough in an oiled container (these are awesome). Put the container in a warm place until the dough has doubled.
Once your dough has doubled, gently remove it from the container onto a lightly floured surface. Stretch and fold your dough (in thirds, like a letter). Then rotate and do it again. Gently return the dough to your container for a second rise.
Let your dough rise until doubled again (I use a crayon to mark the side of my container so I know exactly where double should be). If your dough has been pretty active so far (if your first rise took less than an hour to double), I would do the second rise in the refrigerator.
Gently remove the dough from your container, onto a lightly floured surface and gently shape your loaf.  (One option is the batard.)
Place the shaped loaf on a lightly greased cookie sheet, on a SilPain (I love mine thoroughly), or on a nice layer of cornmeal on your peel, ready to slide directly onto the stone.  Cover the dough. (I use either loosely tented plastic wrap, or a clean dishtowel.)
Turn on your oven. Set the temperature 10% higher than the recipe calls for.(I preheat my oven during all of final proofing, with my bread stone on the lowest rack so it will be nice and hot. If you haven't tried baking with a stone, I cannot recommend it too highly.  The higher temperature is so your bread will have a nice initial spring in the oven.)
When your loaf has nearly doubled in size, mist it with water and slash it several times. (If you did your final shaped rise in the refrigerator, you don't need to let it sit at room temperature before putting it the oven.)
Get a cup full of ice cubes ready.
Open your oven, place the loaf on the stone, and drop the ice cubes on the floor of your oven (If you care about water stains in the bottom of your oven, you can place a cookie sheet or cast-iron skillet in the bottom to catch the ice cubes. If you do, put it in the oven before preheating.) Quickly, but gently close the door.
After 5 minutes, reduce your oven temperature to the normal baking temperature. 
After another 10 minutes, crack the oven door briefly to release the steam.
A loaf like this will be done when the internal temperature reaches about 200˚F.
Remove the loaf to a cooling rack.  Resist the temptation to immediately devour it, as the texture of the crumb and the tenderness of the crust will be adversely effected if you cut it before the loaf has cooled somewhat.


Some resources that have helped me in my bread baking:


The Bread Bible
King Arthur Flour
Using a kitchen scale to measure by weight, rather than volume. (I understand the desire to throw a loaf together, eyeballing the ingredients, kneading with your hands.  It seems as though something as organic as creating bread should be like that. Every time I have tried, my loaf has been inferior. I have been baking bread for 20 years and am sad to admit that if I want my bread to be amazing, I have to measure everything precisely and use a mixer.)