In the Pottermore article on Platform 9 and 3/4 it's mentioned that other fractional platforms exist (7 1/2, for instance). Is it ever stated elsewhere where these platforms lead? Perhaps you could get a connection from your hometown by accessing one of these platforms?
Dumbledore smiled at him. ‘We are in King’s Cross, you say? I think that if you decided not to go back, you would be able to ... let’s say ... board a train.’
‘And where would it take me?’
‘On,’ said Dumbledore simply.
Deathly Hallows - page 578 - UK - chapter 35, King's Cross
Perhaps the ¼ and ½ platforms simply lead On . . . wherever and whatever On may be!
Barring Word of God, I think this is the closest canon reference we're going to get to the other trains. Yes, I realize Dumbledore is speaking metaphorically. And yes, I realize the trains are not specifically described in canon.
On is just an idea. :)
ETA: This kept niggling at me, so I signed into Pottermore and checked the information on Platform 9¾ . . . and I found something. JKR perceived other trains leaving from other fractioned platforms. She thought one might be like the Orient Express, while others would be used on special occasions to take witches and wizards to one-off events, like a concert (or maybe even a magical creatures zoo or circus!)
In choosing the number of the concealed platform that would take young witches and wizards to boarding school, I decided that it would have to be a number between those of the Muggle platforms - therefore, it was clearly a fraction. This raised the interesting question of how many other fractional platforms lay between the whole-numbered platforms at King's Cross, and I concluded that were probably quite a few. Although these are never mentioned in the book, I like to think that it is possible to take a version of the Orient Express off to wizard-only villages in continental Europe (try platform seven and a half), and that other platforms may be opened on an as-required-basis, for instance for large, one-off events such as Celestina Warbeck concerts (see your ticket for details).
The number nine and three-quarters presented itself without much conscious thought, and I liked it so much that I took it at once. It is the 'three-quarters' that makes it, of course.
[J.K. ROWLING - POTTERMORE - PLATFORM 9¾]