Complex sentence: Verizon continues to use the Bell logo on its payphones (including former GTE payphones), hard hats, trucks, and buildings likely intending to display continued use in order to maintain the company's trademark rights.
Split into simpler sentences: Verizon continues to use the Bell logo on its payphones (including former GTE payphones), hard hats, trucks, and buildings. In both cases, these are likely intended to display continued use in order to maintain the companies' respective trademark rights.