Welcome to the POA. What kind of LSA do you have?
To your question. No, you can fly VFR in E airspace under C and B shelves all day long. But if you're flying X Country through that area, why would you? Ask for Flight Following either from your friendly airport on ground frequency before you taxi and see if they can arrange it, or if they can't when you're airborne simply call up approach/departure/whatever ATC you have in your area and ask. HIGHLY recommend it for any X Country.
BTW - If you go under the shelf of any B airspace, while not in the B airspace, you are in their Mode C Veil, and you need a Mode C transponder.
Mode C veil refers to a kind of airspace which currently surrounds all primary Class B airports within the United States. This airspace extends horizontally to a circle of 30 NM radius centered on the airport, and extends vertically from the surface up to 10,000 feet MSL.