Need to migrate from 5.4.2 -> 5.6.1 manually - DB questions

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My site hosting company was (un)kind enough to upgrade PHP to a newer version incompatible with my 5.4.2 C5 installation without telling me. Since I can't access the C5 site anymore, I can't upgrade easily. I'm left with copying my DB and files (which I've done) and installing 5.6.1.

Now I need to migrate the content from my old 5.4.2 DB to the new 5.6.1 DB. Trouble is, the DB structure has changed, and I'm not sure which tables I need for this to succeed.

Does anyone know if this can be done fairly painlessly? If not, I may be left with redesigning the site from the ground-up (most likely with another hosting company - one that notifies their clients before upgrading server-side web services).