بسم الله الرحمن الرحيم
No! There is no need to do ghusl again after performing it on the said pattern. You may offer salah without wudhu after ghusl.
Note: But one shall not get the reward of wudhu in this case. However, if he did wudhu after gargling and entering water into the nose as well as poured water all over his body then he shall get the reward of wudhu.
Allah knows Best!
Darul Uloom Deoband