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Provided you are the sole owner of the business, you should simply register the new business name and start trading under the new name for the whole business or whatever aspect of the business you choose.
If you are in a partnership, or operating through a company with other shareholders, then you will need to seek the approval of the other parties involved first.
I trust the above assists.
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You can treat all your businesses as one, or you can have multiple business names to reflect the diversity of your business operations. The later is usually best for marketing reasons, but legally you don't have to have different business names for every business activity you conduct.
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You are obliged to register any business name you use with the public, and therefore if you wish to use a different business name for each of your business activities, they must each be registered.
I suspect you aren't doing the search correctly. Organisation and business name searches produce many pages of Jims related business names. Jim's Group is a corporation, of which there are many fewer than business names.