https://stackblitz.com/edit/react-hogvhv?file=index.js

I have tested this and it works according to what you requested.

import React, { Component } from 'react';

import { render } from 'react-dom';

import './style.css';

import DatePicker from "react-datepicker";

import "react-datepicker/dist/react-datepicker.css";

import moment from 'moment';





class App extends Component {

  constructor() {

    super();

    this.state = {

      selected: new Date(),

      minTime: this.calculateMinTime(new Date())

    };

  }

  calculateMinTime = date => {

    let isToday = moment(date).isSame(moment(), 'day');

    if (isToday) {

        let nowAddOneHour = moment(new Date()).add({hours: 1}).toDate();

        return nowAddOneHour;

    }

    return moment().startOf('day').toDate(); 

}



  handle = (date) => {

    this.setState({

      selectedDate: date,

      minTime: this.calculateMinTime(date)

    })

  }



  render() {

    return (

      <div>

         <DatePicker

            selected={this.state.selected}

            onChange={this.handle}

            excludeOut

            showTimeSelect

            timeFormat="HH:mm"

            timeIntervals={15}

            dateFormat="MMMM d, yyyy h:mm aa"

            timeCaption="time"

            minDate={new Date()}

            minTime={this.state.minTime}

            maxTime={moment().endOf('day').toDate()}

          />

      </div>

    );

  }

}



render(<App />, document.getElementById('root'));