If it's running the IOS switch code and not the CatOS code you might try the Backup command in the interface config. you can specify a backup interface that will become active if that link becomes unactive. Or if both switches support trunking you could setup a trunk between them.
The Catalyst 3550 can be used as Layer2/Layer3 device. I don't know how you are setting the switch up, but I am assuming that it is L2--all ports on the switch belong to the same VLAN. In that case, if the other switch (non-Cisco) supports Spanning-tree standard, then by just connecting the switches together and let Spanning-tree figures out which path to block.
We are pleased to announce availability of Beta software for 16.6.3. 16.6.3 will be the second rebuild on the 16.6 release train targeted towards Catalyst 9500/9400/9300/3850/3650 switching platforms. We are looking for early feedback from custome...