Hi,
If you want to do EE in a design firm I would look at the larger firms that work at the forefront of innovation and experiment with new technologies. If you have a sense for design, are good at working in teams doing rapid development cycles in semi-chaotic environments with tight deadlines and you have good ideas about innovation then you have a good chance in being able to merge the two fields. More important than your resume and school achievements is your portfolio, also as an engineer. Focus on upcoming technologies, you are likely to land jobs if you have knowledge and skills in specific fields such as software for intelligent robots, autonomous cars, drones, 3D printer firmware etc.
-Ralph