If it means everything will work the way it does as you are now shooting with only the EF-S 55-250mm f/4-5.6 lens the answer is no. Autofocus: Because your T3 limits your auto focus system to lenses with maximum apertures of f/5.6 or wider, even a 1.4x teleconverter will disable your AF system when using your EF-S 55-250mm f/4-5.6 lens.
Choice of Canon 55-200mm f/4.0-5.6 USM 55-250mm IS EF EFS STM Auto Focus Lens. $138.28. Free shipping. eBay Refurbished.
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canon ef s 55 250mm photo examples