How Can I Legally Improve a Product I Didn't Create?
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I'm assuming from your question that when you refer to "the company," you are referring to your employer. So, can you profit from your employer’s pre-existing intellectual property? That depends on your employer. Some companies are excited about encouraging innovation in their employees. They may even have a written policy of some kind of revenue share for new ideas that employees introduce to the company that are implemented.
Many, however, take the opposite approach. In their view, anything that an employee creates on-the-job belongs to the employer. After all, the company is giving you a job and paying your salary. In exchange, you are providing your brainpower, loyalty and effort (at least in theory). Copyright law certainly supports this perspective. "Work for hire" rules clearly state that the employer (and not the employee) is deemed the author of any works created by an employee within the scope of his or her employment. In addition, if the product is protected under a patent, the patent holder (whom I am assuming is not you) has the right to exclude others from making, using or selling the product.
If you're serious about pursuing this, absolutely speak to a business attorney to (1) review any employee manual or contact you may have signed that covers ownership of inventions created during the course of your employment, (2) determine any quirks under your state's employment laws that might affect your situation and (3) get guidance on whether your proposed idea would infringe on your employer’s intellectual property.
And even if "the company" is not your employer, you'll still want to speak to an intellectual property attorney to explore whether your proposed idea would infringe on that company’s intellectual property.