Dividing up Hexagonal Board Tiles
consider looking at how other games solve this problem, For example Settlers of Catan adds a "sea" around the boad which fills in the uneven edges when using full hexagons and gives a uniform diamond at the edge. Something along those lines might be the best way to resolve your problem.

Alternatively you can design your overall boards such that the overflow hex tiles "jigsaw" together, either each board has half tiles which will always match up well (the style and substance of your tiles is important in making this decision, if you have an aliens vs robots theme for example you cant have tiles that are 50% alien 50% robot) or having some of the half tiles whole, some empty to slot together, which will also have a positive impact on the feel of your boards since they wont slide about as easily.