Q: Hi is there a way to continue with this trademark process?

A: If you had an application in progress to register the mark, and it was not completed, it cannot be revived.

Since you refer to both a "logo" and a "name", I infer that the mark had both a text component and a design component. It is possible to register just the text mark, or the combined text and design.

You should file a new application to register either the text mark alone, or the text with the new design component.

A: I would just like to add that before you start a new registration application, you must conduct a common law clearance search for your mark, covering every single U.S. geographic market. The Lanham Act does grant a presumption of exclusive use throughout the territorial US to the first successful registrant. Yet, common law trademark rights vest in the first user of the mark in every geographical market segment, whether or not the mark is registered. For this reason, some people find that after they successfully register their mark with USPTO and start selling their product under the brand, they get hit with a lawsuit from a concurrent brand user claiming rights to a related market in some part of the U.S. The legal fees and distraction from such cases, not to mention the years they take to resolve, can easily bankrupt a startup as you are barely getting started. This can all be potentially avoided with a competently conducted common law brand study and search. If the brand has potential value, it borders on incompetence not to conduct your full IP due diligence before sinking your resources in a product line that can quickly turn into a total loss, regardless of the inherent value of the product you are bringing to market.