The majority of agents who went in by parachute, especially later on in the war, would have been met by a reception committee, meaning that they would not need to immediately blend in with the local population. They would be greeted on the ground by a party of helpers, and as long as they hadn't had the misfortune to be landed into a German trap as happened with some agents, their first night would be spent at a nearby house before travelling on to their destinations. This would mean they didn't have to immediately change upon landing.

SOE did not just parachute in agents. There were a number of insertions by boat, particularly in the south of France which was beyond the range of British planes until later in the war, and many agents were inserted into France via plane, but the plane would land to allow them to disembark by landing. This also allowed returning agents to be picked up without having to trek over the Pyrenees or make their way to the coast for a seaborne extraction. In this case, they could go into the field wearing a normal outfit.

Agents would often take a whole wardrobe of clothes with them, to prevent them having to either source additional clothes in the field or be given extras by the networks they joined. Violette Szabo for instance took two large suitcases with her and had to cycle to the house she was due to spend the night with one of them strapped to her back, although she was landed by plane and not parachuted so could take more luggage with her. These clothes were either adapted from British wardrobes by a team of tailors - removing and replacing labels, linings, buttons and laundry marks - repurposed from returning agents or sourced from French refugees arriving in the UK. This also extended to toiletries and personal effects - if it would be out of place in France it would be out of the question to take.

Before agents got on to the plane to leave, they would often go through one final check to ensure no incriminating English train tickets or small change remained. However, this was not foolproof. John Farmer, who parachuted in alongside Nancy Wake, discovered a British regimental badge in his pocket despite having been searched shortly before leaving. Had a search by German soldiers or French police discovered this, it would have been very difficult to explain away.

What an agent was wearing tends not to be the kind of thing that gets recorded in official sources, but as far as I can tell there was no hard and fast rule as to what an agent would wear. From a quick search through the books I have to hand I can only find one outfit description. Russell Braddon's biography of Wake describes her outfit thus: "smart civilian outfit [not specified whether skirt or trousers], silk stockings and three-quarter-heeled shoes. On top of them, for warmth and pocket room, were a pair of overalls". Wake was wearing a camel-hair coat on top of that. As you rightly suggest, carrying out a parachute landing in heels is not the most comfortable experience so her ankles were heavily bandaged to alleviate the shock of landing. In the end, the bandaging of her ankles was unecessary as she was blown off course and got stuck in a tree.

Parachutes were supposed to be buried near the landing site, but the appearance of so much silk in wartime, heavily-rationed France, meant that sometimes they were repurposed to make new clothing. I can't find any refence as to what happened to overalls but it seems likely they would suffer the same fate as the parachutes.