Your property taxes and their increases are driven primarily by the revenue each taxing entity requires (the whole pie).
The tax appraisal is not the cause of increased tax revenue needs, it is instead the knife which slices your share of the whole pie.
Tax protests only help if your property is inequitably appraised. You have a window to file your protest between May and July, depending on when the appraisal districts send your appraisal notice.
In short, the best way to trim your tax payments is to let your voice be heard during city, county and school district elections and during budget hearings. Once you hear the cases for and against the proposed budget, you can make a decision whether or not the accept the budget as valid and necessary or speak against excessive spending and tax increases.
It's a simple concept yet it is usually clouded by political rhetoric and obfuscation of facts (i.e. "but our tax rate has remained the same, our tax rate is one of the lowest in the state, etc.)