I found a connector that might work - this one here: 64 Way 2 Row Connector SMT 1mm Pitch AMP 120521-1 2 PIECES Z307
I bought some to play around with and the pin pitch is correct. The connector must be trimmed down to fit in a rather delicate way - each of those pins is ~1mm:
The attachment Toshiba-T4900CT-makeshift-connector2 - Copy.jpg is no longer available
But it did fit onto the connector with some wiggling.
I did have continuity on the pins I checked, with none shorting pins to adjacent pins.
The attachment Toshiba-T4900CT-makeshift-connector1 - Copy.jpg is no longer available
This connector is too tall to make a PCB fit onto directly, which was my original hope - but the cover does still close with this connector in place. I was going to suggest that you could get this and solder wires on for all the pins, but I've got a better idea - what if I make a PCB that solders onto the upper part of the pins facing the HD bay, then run a cheap and easily available flex cable onto pads on that PCB, like this:
The attachment T4900CT-designidea.png is no longer available
Or maybe even simpler, just make a PCB that will be soldered onto two strips of 6cm flex cable, then perhaps that could even just have a regular 44 pin IDE connector