In short, the answer is, under normal circumstances, never. You shouldn't bother to get a visitor visa unless you have had some iimmigration problem in the past, and you should not apply to extend your stay in the UK unless you want to stay past the date that was stamped in your passport when you entered.
An extended visa is for someone who needs a visa to be in the UK, and who needs to stay in the UK beyond the expiration of the visa. You don't need a visa, so you don't need to extend a visa. If you plan to spend a total of less than six months in the UK, but spread out over a period of seven months, there should be no problem.
Each entry to the UK is a new entry, and the normal course of affairs is that you get six months on each entry. In other words, your six-month eligibility is automatically broken up when you leave the country. If the immigration officer notices that you've spent a good deal of time in the UK recently, they may admit you for less than six months, but the time should be reduced by your recent stay in the UK, so the total will still be six months.
For example, if you stay in the UK for four months from November through February, then enter at the beginning of April for two months, you are likely either to be given six months or two months.
Extending a vistor visa costs £811, which is around $1050 US dollars right now. It's a very expensive undertaking for something that is not actually necessary.