You're smart to be looking to redesign your site to enhance search results. If your site is holding you back, you’ll likely need a site redesign before you can achieve good SEO results. Search Engine Optimization should start at the very beginning of a website. Trying to optimize a poorly designed site is a time consuming, inefficient way to go about internet marketing, and it can often yield very poor results.
I agree about adding an "About" section to establish the company. Maybe you could add social share buttons to your blog postings and latest news sections to gain more engagement.
If you want to complete a site review on your own try using SEOzio. Its a free tool that provides a fast and easy way for people to see how their website performs from an SEO perspective. You can look at individual parts of the SEO puzzle like link analysis, branding, crawl-ability of the site and site code, and how each factor contributes to you website's overall score.
Good luck!