Can I use 'better still' in negative sentences?

Solution 1:

As an aside, I think three articles can be dropped to form a clearer sentence. Also, the question mark confuses me a little bit, because the sentence is structured as a factual statement, not as a question. You can of course add a question mark to turn any sentence into a question, using intonation, but I'll assume a statement for now.

You may not have access to trusted counselling, or better still, to 24/7 support.

The obvious problem is that it's not clear that access to 24/7 support is also affected by the negation, because of the semantic clash with better. I would suggest to use another option, to indicate that 24/7 support is even harder to find, rather than much better to have:

You may not have access to trusted counselling, let alone 24/7 support.

As Merriam-Webster says for let alone:

to say nothing of : not to mention —used especially to emphasize the improbability of a contrasting example
“he would never walk again let alone play golf — Sports Illus.”
“how many ever see an Ambassador or Minister, let alone a President — Robert Lacville”

Solution 2:

I believe the "negative" version you're looking for is worse yet:

You may not have access to trusted counseling, worse yet 24/7 support?

Alternately or worse:

You may not have access to trusted counseling, or worse: 24/7 support?

Or you could reverse the order to make use of much less:

You may not have access to 24/7 support, much less trusted counseling?

The order must be reversed in this last structure so that the scope of the former encompasses the latter.