My guessing game works, but I'm sure the code could be improved. Any suggestions?

import random

def main_game():
  print("You have 5 guesses!")
  rand_num = random.randint(1,10)
  guess_count = 5
  while True:
    guess = input("Pick a number between 1 and 10!: ")
    try:
      user_num = int(guess)
    except:
      print("That wasn't a number! Try again. You have {} guesses.".format(guess_count))
      continue
    guess_count = guess_count - 1
    if user_num == rand_num:
      print("Great job! The number was {}. You got it in {} guesses!".format(rand_num,5-guess_count))
      play_again()
    elif guess_count == 0:
      print("You ran out of guesses! The number was {}. Better luck next time!".format(rand_num))
      play_again()
    elif user_num >= rand_num:
      print("Too high! You have {} more guesses. Guess again!".format(guess_count))
      continue
    elif user_num <= rand_num:
      print("Too low! You have {} more guesses. Guess again!".format(guess_count))
      continue

def play_again():
  print("Thank you for playing! Type 'P' to play again.")
  again = input("> ")
  if again == "P":
    main_game()
  elif again != "P":
    print("Command not recognized.")
    play_again()

main_game()
play_again()