Its almost impossible to guess an insect id without a very detailed description, or a photo.

A couple of ideas on places to start:
What seeds are in the pots and where are they being kept?
Are there any pots they are not in, and what is sown in those?

Do you have a camera with a good macro? If so, try and put a couple of the insects into a zip-lock and stick it in your fridge for about 10 minutes. It should chill them but not kill them (or keep 'em in longer to kill them if you like) and render them very nearly motionless. Now put them on something white, snap some good macro pics, and post. =)

Perhaps springtails? They infest overly wet places, are very small, come in different colors, and jump.

Tried taking a picture, but they're just too small to get a decent focus on. They really do look like spider mites in that they have a dark body but tiny lightly colored (legs?) sticking out in front but slightly larger in overall size. They also have antennae like aphids. Smooth, solid black with a tiny white speck on the left and right sides. Their undersides are white-ish. They tend to hop when disturbed or trying to get to something and they're pretty good at it. I'm only finding them in the top 1/16" of soil, but I've also noticed a few green aphids hanging around too on the surface. They seem to be spreading to other containers.

So far, I've found them in containers of seedlings/seeds of:
Texas bluebonnet
Platycodon
Aster
Datura (more recent)
Blackberry Lily
All containers have a bit of algae growth in them.

Overnight, I noticed that they flocked to a cup of water that I had a chrysanthemum cutting in it. Most drowned, but several seem to be hanging on to the stem a few inches above the water, which leads me to think that they're an aphid or mite of some sort. Once again I noticed another green aphid nearby.

You wrote: "I noticed that they flocked to a cup of water ... ... several seem to be hanging on to the stem a few inches above the water, which leads me to think that they're an aphid or mite of some sort."

Neither mites nor aphids do that.

Try taking a picture on your macro setting. Depending upon what sort of camera you have, you can take a clear image from between 2 & 4 inches away from the subject. Then post full size image. Or crop.

I'll try that when I get home. Here's what I found after an extensive search. Finally! It's exactly what this critter looks like. You were right, it is a springtail... a globular springtail. I've never seen or heard of these before. Now to figure out how to rid the hoop-house of them.

Picture 1
Picture 2

As was said, springtails do well in moist places. Dry the hoop house out. Fans? Ventilation?

No fans or ventilation, but it is time to open it. The dirt in it is dry. The containers of seeds and seedlings are having the problem, and letting those dry out completely isn't an option.

I've read about watering affected containers with bleach, vinegar, or malathion solutions, but I'm a bit hesitant since the seedlings are so young and tender.

you said: "seedings ...letting those dry out completely isn't an option"

Of course not. But perhaps you can back off a bit.

Even so, high humidity also enhances springtail habitat.

And when you have an enclosed area such as a hoop house, you absolutely must have some means of ventilation.