When designing a breakout board, it's a good idea to oversize the 14 holes for the interior connections. On one I made for testing, I made the 28 outside holes 0.038 inch, which is just barely large enough for a normal header pin. On the interior holes, I made them 0.050 diameter.
When you assemble the thing, you first need to solder a 14 pin J-lead header to the Teensy3. It will not align perfectly. The larger holes allow for a small misalignment.
To complete the assembly, the easiest way involves putting the two header strips into the 0.038 holes with the short side facing up. Then place the Teensy3 onto the board. The 7x2 header that's already soldered to the Teensy3 will mate with the 0.050 holes first. Then because the two header strips on the outside are not soldered, you can wiggle the Teensy3 into position onto the 2 loose header strips. If your placement of the 7x2 header was highly accurate, everything will be a loose fit. If it's off a little, hopefully this way gives you the best chance get things lined up and press the Teensy3 into position with all 28 other pins.
Also, you might consider whether you really truly need any of those bottom-side signals. Some projects really do need a lot of I/O. I know it's really painful to have a chip with so much I/O but no way to access those other pins (eg, Arduino's larger Mega & Due have much inaccessible pins). But for most projects, the main pins on the top side are plenty, and they're much easier to access.
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