[dice]

Maybe you could make newspaper pots out of
a tennis ball can. That makes kind of a tall,
narrow pot with about a 3" diameter, plenty
of room for the roots until plant out but one
can still fit lots of them under 4' flourescent
lights.

I have noticed that newspaper pots dry out
more quickly than plastic pots (the newspaper
absorbs water from the soil and allows it to
evaporate from a larger surface area), but that
is not really a liability with tomato seedlings,
where the opposite problem, drying out too
slowly, is more of an issue for plant health
and root development. One simply must
keep a closer eye on them in newspaper pots
than in plastic pots. I can usually tell by the
weight of a newspaper pot in my hand when
it needs water. (One can keep one the same size
filled with dry planting mix and no plant in it
nearby for comparison.)

Not being an origami wizard, I found that a piece
of duct tape on the folded bottom and a couple
of rubber bands simplifies newspaper pot construction
and potting up into them. It is wonderfully easy
to remove the root ball intact, without root damage,
when it is time to plant out or pot up to a bigger
size pot. One simply removes the duct tape, unfolds
the bottom, and carefully unrolls the newspaper
until the root ball rolls out into your hand.

If a tennis ball can seems ridiculously tall for the
amount of time that you have until plant out,
maybe only use 2/3 of the height of the tennis ball
can when making the newspaper pots. That still
keeps them narrow enough to fit many under lights
or on a window sill.