Tax on Settlement Agreement

[anonman123]

I was let go from work on a 'Settlement agreement' which basically means that they would pay me 6 months worth of my annual salary. I have a question regarding how it was taxed and want to know if I am due any tax. My financial details are as below:

March 2024 - Received final pay (seemed ok)
April 2024 - Received settlement payment

My salary was £44k, which means I would be paid £22k minus any taxes. The amount I actually received in April was £11,268.38

My take-home pay for the months prior was £2,592 per month. I assumed my take-home pay from the settlement would be 6 times this, which would mean I should be paid around £15,552.

Then I got a new job in May 2024. I informed HMRC and they said I would be taxed more going forwards because the £22k would be treated as additional to my current income at my new employer. I understand this, but I thought I should be receiving a tax rebate because I believe I was initially over taxed.

Could someone clarify if I was over-taxed or if the amount I received looks correct?