The answer is unusually specific, because Plath spelled it out herself. In the last months of her life she wrote before dawn, in the one stretch of the day that belonged to no one else, and that hour produced most of the work she is remembered for. It is one of the most precisely documented routines in modern poetry.

The Direct Answer
At about four in the morning, before her two children woke.
She described the hour herself in prefatory remarks written for the BBC in late 1962.

The Hour Before the House Woke

By the autumn of 1962, Plath was living at Court Green, a thatched farmhouse in North Tawton, Devon, and raising two small children largely on her own after her marriage to Ted Hughes fell apart. Her son Nicholas was still a baby and her daughter Frieda had just turned two. The daylight hours belonged to them, to feeding and washing and the endless upkeep of a house run single-handed. So Plath wrote in the only hours that were truly hers, the ones before anyone else was awake.

She left an unusually exact record of when that was. Introducing her new poems for a BBC broadcast late that year, she pinned the time down herself.

These new poems of mine have one thing in common. They were all written at about four in the morning, that still, blue, almost eternal hour before cockcrow, before the baby's cry, before the glassy music of the milkman, settling his bottles.

Sylvia Plath, prefatory remarks for the BBC, late 1962

A letter to her mother that October puts a slightly earlier clock on the same habit. She was up 'about five, in my study with coffee, writing like mad,' she wrote, and had 'managed a poem a day before breakfast.' Four o'clock or five, the picture holds. She woke in the dark, made coffee, and worked until the household stirred.

What the Quiet Hour Produced

This was a late habit, not a lifelong one. For years Plath had written in ordinary daytime hours, often fighting hard to get anything down and revising herself into knots. In Devon, cut down to a few uninterrupted hours before sunrise, she wrote faster and more freely than at any point in her life. October 1962 alone gave her around twenty-five poems, including Lady Lazarus and Daddy. She wrote the title poem, Ariel, in the predawn of October 27, her thirtieth birthday, and finished a second poem, Poppies in October, the same morning.

The pull of that hour was more than the quiet. It was a sealed room in time, closed off before the world's demands could reach the page. Once Frieda and Nicholas woke, Plath was a mother again for the rest of the day, and the writing had to be done before the first cry from the nursery.

The sleeping pill she mentioned belongs in the picture too. Plath was under genuine strain that winter and taking medication to sleep, and it tended to wear off in the small hours and wake her. She turned that early, broken waking into working time. The routine reads best as a hard discipline pressed out of a hard season, rather than a tidy productivity trick to copy wholesale.

See the full Sylvia Plath writing routine for how the study, the journals, and the Devon months fit together.

What You Can Steal