Board & Card Games Stack Exchange is a question and answer site for people who like playing board games, designing board games or modifying the rules of existing board games. It only takes a minute to sign up.
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.
Mage Knight: The Board Game uses a series of boards that are made of smaller hexes. Each board is 7 hexes and connects to the adjacent board. There are also symbols along the edges which dictate the orientation of each board.
Hexagons are naturally dividable into 6 equilateral triangles.
Or you could opt for other than hexagonal boards. If you want hexagonal tiles, you could have collections of 7 (a central one with 6 surrounding ones), 10 (2 central ones with 8 surrounding ones), 12 (3 central ones with 9 surrounding ones), 14 (4 central ones with 10 surrounding ones) or 19 (7 central ones with 12 surrounding ones) tiles in a (sub)board.
Doing a comparison on how other games do it is your best option. However, don't limit yourself to just other board games. The first game that comes to mind is Heroes of Might and Magic 3. You should be able to see how they constructed 'boards' in that.
Like @Patters said, Catan is a great example. Hecatomb is a card game that leveraged odd shaped cards and Hive does a great example of moving around 6-sided pieces.