VUM is a part/component of vCenter server installation you cannot manage ESXi 5.5 hosts with VUM 6 without vCenter 6 in your environment ... (can be at the same machine as VC or on diff comp)
And in addition always start with vCenter server upgrade...follow order below but before have a look at these guides:
http://kb.vmware.com/selfservice/microsites/search.do?language=en_US&cmd=displayKC&externalId=210977...
http://kb.vmware.com/selfservice/microsites/search.do?language=en_US&cmd=displayKC&externalId=210854...
- Upgrade First vCenter Single Sign On
- VMware vCenter Inventory Service
- VMware vCenter Server
- VMware vSphere Client
- VMware vSphere Web Client
- VMware vSphere Update Manager
- Upgrade ESXi hosts (VUM)
- Upgrade VMTools (VUM)
- Upgrade Virtual Hardware (VUM)
- Upgrade Product licenses (vCenter , ESXi, …)
- Upgrade VMFS to proper version (if applicable)
- Upgrade SRM (if deployed)
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Regards,
P.