Still wanting to run eSATA on the external enclosure, I decided to purchase another eSATA card, but this time based on a Marvell 88SE9123 chipset instead. At the same time, thinking that it could be a cable issue, got 2 eSATA cables together from Amazon.
Installation of the card was relatively painless. But the external enclosure was not being recognised properly so it was back to the SIL3132.
The new cables were installed and a few days went by without any issues. So it might be the cables after all.