Использование собственных методов
Проверьте эту песочницу: https://codesandbox.io/s/flatpicker-with-current-time-6tg7j
<Flatpickr
ref={datepickerRef}
options={options}
value={""}
onChange={(date, dateStr) => {
let selectedDate = date[0];
let currentDate = new Date();
let currentTime = new Date(
selectedDate.setHours(
currentDate.getHours(),
currentDate.getMinutes(),
currentDate.getSeconds()
)
);
setDate(currentTime);
}}
/>