Texas Lawyer's Adrian Dayton wrote a really great article recently on how to build a list of your ideal clients while gaining insights into how best to connect with these hard to reach people. The first step is to find them.
Start by selecting advanced search. Enter the title of the people you are trying to reach, such as "General Counsel". adding the ZIP code and set geographical search limit for 50 miles. The result will be a treasure trove of information. Not only do you get names, but context. Connections and people in common. The most important information garnered in these searches are the people you have in common with your target audience. Pick up the phone, call your connected person and ask them to introduce you.
Invite your targeted person to a CLE, or ask them industry-specific questions for an article you are writing. You can also use Linkedin's "get introduced" feature; this allows your common connection to forward a message form you to your target.