If I currently hold a Secret clearance and apply for TS jobs, will my application be automatically denied? Or if I meet their technical requirements and what they prefer in an applicant, is there a chance they may contact me even though I don’t meet the TS requirement?
If the employer is willing to wait then it will not be a problem. But a lot of employers will skip over your application if their first concern is to get someone on a contract ASAP.
If a company also has work at the Secret level and they don’t have any organizational barriers to prevent you from working on them (and they do exist, trust me) then they may be more likely to extend an offer.
I agree with @sbusquirrel , if the position requires you to hit the ground running with access to TS information/space, then you won’t get the job. If they have some room where you can do other work with restrictions (i.e. no TS info or space-access), or (uncommon) can put you on another contract at the Secret level, then your experience/skills may be enough to off-set the lack of active TS.
I’ve worked contracts where both have happened: you must have the required clearance level as of day 1, and others that you’ll need it but work can be done at a lower level while you’re processed for a higher clearance.
As @sbusquirrel said, it depends on the employer.
Sometimes the employer will indicate their preference on the job description in the ad. If it says “ability to obtain/maintain Top Secret clearance”, that means they are willing to wait. If it says something like “Current Top Secret clearance required”, that probably means they will skip over you if you don’t already have it.