Discrete Schmitt Trigger: make output logic level at 0V, not 3V
The basic problem is how the positive feedback in your Schmitt trigger robs low output voltage.  The feedback voltage is generated across RE, which is always added to the low level output voltage.

A Schmitt trigger is basically a amplifier with a little bit of positive DC feedback.  Can you think of other ways of using two transistors to make positive gain?  Perhaps some of those configurations allow for a feedback path that does not limit the output voltage swing.

Hint 1: Your exising amplifier topology can be used with a different feedback path.

Hint 2: The two transistors can be a mix of NPN and PNP.  That may open your thinking to more amplifier topologies.