As others have pointed out, the CLUE database is a pain ... so including some of the personal items on another policy from the house might allow you to make claims without risking the homeowner's policy and putting you in a bind with the mortgage company.
So yes, looking at supplemental insurance might seem expensive, but be relatively cheap if you ever need to make a claim.
Sort of sucks with CLUE that even talking to your agent about a "hypothetical" claim can put the policy at risk these days.