Sleep for more than 1 hour using EEPROM to keep the count. Technically, this is not sleeping for more than 20 minutes, the maximum possible deep sleep, but it will keep a count every time it wakes up so that at the correct count, it will possibly enable the radio and do a desired task.
If only ESP8266 is used, short pin D0 to pin RST to enable wakeup.
#include <EEPROM.h>
const int sleepTime = 1200;  // 1200 seconds = 20 minutes
#define FINAL_SLEEP_INTERVAL 6  // 6 * sleepTime = 6 hours
#define CURRENT_SLEEP_INTERVAL_ADDR 30  // Store sleep interval
#define CURRENT_SLEEP_INTERVAL EEPROM.read(CURRENT_SLEEP_INTERVAL_ADDR)
void setup() {
  EEPROM.begin(512);
  Serial.begin(115200);
  pinMode(2, OUTPUT);
  Serial.println("[INFO] Wake up!");
  if (CURRENT_SLEEP_INTERVAL == FINAL_SLEEP_INTERVAL) {
    blink(5);
    resetSleepInterval();
    doSomething();
    goToSleep();
  } else {
    blink(5);
    increaseSleepInterval();
    goToSleep();
  }
}
void blink(int times) {
  for (int i=0; i <= times; i++) {
    digitalWrite(2, HIGH);
    delay(1000);
    digitalWrite(2, LOW);
    delay(1000);
  }
}
void loop() {}
void doSomething() {
  Serial.println("[INFO] Hey do something finally :)");
}
void goToSleep() {
  ESP.deepSleep(sleepTime * 1000000);
}
void resetSleepInterval() {
  EEPROM.write(CURRENT_SLEEP_INTERVAL_ADDR, 1);
  EEPROM.commit();
}
void increaseSleepInterval() {
  EEPROM.write(CURRENT_SLEEP_INTERVAL_ADDR, CURRENT_SLEEP_INTERVAL + 1);
  EEPROM.commit();
}.PHONY: lint compile upload clean
lint:
	cpplint --extensions=ino --filter=-legal/copyright *.ino
compile:
	arduino-cli compile --fqbn esp8266com:esp8266:d1_mini ./
upload:
	arduino-cli upload -p /dev/cu.wchusbserial1410 --fqbn esp8266com:esp8266:d1_mini ./
clean:
	rm -f .*.bin
	rm -f .*.elf
flash: lint compile upload clean