English is so rich yet so weird sometimes.
I just Googled this and really hadn't thought much about it but when the following SOUND is like a consonant, you say thuh, e.g., the university. Thuh house but thee hour.

if thou art a native English speaker, thou hast followed this rule thy whole life without knowing it.

"thuh hour"

"thee hourly newscast."

We're not talking about citation-form stress, but full stress after resolution of stress clash and stress reduction to show noun phrase boundaries. So "hourly" may officially have initial stress, but in "hourly newscast" the word "hourly" has scant secondary stress. The /i/ in "the" is reduced in any event.

It's a hiatus resolution strategy, to avoid schwa next to a (non-high front) stressless vowel (which tends to be phonetically reduced anyway).

As soon as I say, "no, not the 6 pm newscast, but the HOURly newscast," the schwa is back and anything else sounds weird.

However, as soon as I swap out "hourly" for "evening," "thuh evening news" with stress on "news" is fine. Don't want the two high front vowels in hiatus any more than I want two reduced non-front non-high vowels in hiatus.

Let me see if I have this straight. Schwa is that upside down e character which represents the (non-high front vowel) vowel sound in "uh" or "um". The vowel in "thee" is /i/, which is a high front vowel. Hiatus is two adjacent vowels that don't form a diphthong. Hiatus bothers you most if the two vowel sounds are similar, i.e., both high front or neither high front. Did I get all that right?

I'm confused by your example: "no, not the 6 pm newscast, but the HOURly newscast." Do you say "thuh HOURly ..." or "thee HOURly ..."? In this example you have "the" before a stressed vowel. The title of your post suggests "thee", but you say that the schwa is back and anything else sounds weird.

I suppose there are geographical variations, and maybe individual variations in the vowel sounds of "the" in context. In the past there were masters of "elocution" who taught that unaccented vowels should avoid the tendency toward schwa. That was like trying to push back the tide, but it may have had some residual effect.

in Emily Brontë's novel Wuthering Heights.

According to Wikipedia:

At the very close of the novel, a servant boy tells Nelly that he has seen the ghosts of Heathcliff and Catherine walking the moors together, although Nelly and Lockwood both insist that they must be treated as if their souls were at peace. The novel closes with Lockwood wandering past their graves and wondering "how any one could ever imagine unquiet slumbers for the sleepers in that quiet earth."

read more: http://en.wikipedia.org/wiki/Heathcliff

Another example of this is in "-'ve", it is pronounced "uhv" before a vowel and "uh" before a consonant. Hence "would-a", should-a, and could-a.

My dictionary says:

which seems to cover my topic and a lot more.

I often say "would-uhv", "should-uhv", or "could-uhv" before consonants as well as vowels, but when I am careless the "v" sound is not very loud, i.e., what I say before a consonant is close to "would-uh", "should-uh", or "could-uh".

A common mistake in writing is to replace "have" with "of", e.g., "should of gone". This mistake reflects the pronunciation of "have" in context.