I will be on a 7 day London tour the first week in August. I will have time before the tour starts and on free afternoons during the tour to explore on my own. I need ideas for 1/2 day tours both in London and in outlying areas.
Hampton Court, Kew Gardens, Greenwich, Alexander Palace, Tube to Hampstead, explore the village, meander in the Heath then walk to and explore Camden Lock Market, Chatham Docks, Victoria and Albert Museum, walk along the Thames path, stroll along the Serpentine in Hyde Park ( a meal at the Lido Cafe) ....lots of places to occupy your time pre and post the tour.
Hard to help without knowing what your tour covers. You may want to use those half days for London sights it doesn't cover. The guide might have suggestions based on your interests.
But a few good half-day trips, if they interest you, would be Hampton Court to the west, Greenwich to the east, and Windsor Castle to the west.
Hampton Court (by train from Waterloo) is Henry VIII's palace with major additions by the Stuarts who followed him. We found the Tudor kitchen and associated rooms very interesting. There are also good exhibits about William III. It has a beautiful garden with a maze beside the Thames.
Greenwich (by tube or by Thames boat) was the naval academy, designed by Wren, and still has the excellent naval museum. Uphill is the observatory, which you can visit, and the Greenwich Meridian where longitude is zero. On the river nearby is the Cutty Sark, a famous old clipper ship.
Windsor Castle (train from Paddington) is still used by the royal family but you can tour the elegant state rooms unless something special is happening. The portraits are like a military and political history lesson. St. George's Chapel, a great example of Perpendicular Gothic architecture, has the royal tombs from the 18th century onward. Prince Harry and Megan Markle are getting married there next month.
While not half day, I would jump at some London Walks tours. You might even be able to squeeze in an afternoon and evening walk on the same day when free. Since you don't need reservations nor pre-pay you would not be penalized if for some reason you missed your tour. We enjoyed around a half dozen LW tours while in London last fall. Here's a link to their site: LondonWalks