Impact of old job monthly pay schedule affecting tax liability at new job

I probably have misunderstood. How do I find what the details are of how the new tax has been calculated?

From gov.uk:

Total Income Tax estimate
£20,947

Your total income from employment is estimated at £76,862 this tax year. You have a tax-free amount for your employment income. This means the first £11,850 of your employment earnings are tax free.

Income Tax Band / Amount / Tax Rate / Tax
Tax-free amount / £11,850 / 0% / £0
Basic rate / £35,033 / 20% / £7,006
Higher rate / £29,979 / 40% / £11,991

Additions to your Income Tax

The following all increase the amount of tax you will pay this year

Tax adjustment
Estimate of the tax you owe this year £1,949.60
(I've been saying £2000 as an approximate figure, number is actually £1949.60)

What does this mean?

We think you’ve paid too little Income Tax this year and owe £1,949.60. We’ll start to collect this straight away. We’ll collect £1,949.60 from your taxable income before 5 April 2019. To do this, we’ve reduced the tax-free allowance in your tax code by £8,512 to collect this in equal amounts.

My old job was paying me under tax code 1211L. My first and most recent pay under my new job was under tax code 1185L. HMRC is telling me my tax code is now 360LX.

360
The definition of this part of your tax code is as follows: Your tax-free amount for this employment or pension is £3,600.

L
The definition of this part of your tax code is as follows: You are entitled to the standard tax-free Personal Allowance. Your tax code will be updated automatically if the Personal Allowance changes.

X
The definition of this part of your tax code is as follows: Your tax is based on your pay in each pay period, not the whole year. This stops you paying too much tax in one go. Your payslip or pension statement could show this as W1/M1, or week1/month 1.

Are tax codes affected by pension contributions? If so, my new employer hadn’t activated my pension contributions in time for my first pay in August but they told me it will be in place for September payday. So could it be that my tax code will change again to something more favourable after my first pension contribution this month? As far as HMRC is concerned, there has been no pension deduction in my new job so is their estimate based on that accordingly?

I contributed 2.4% to my pension at my old job and my employer contributed 3%. At my new job, I have opted to contribute 10% while my employer also contributes 10%. These will not be 2.4% and 10% of the gross figures I stated above because my base gross salaries are lower due to other allowances I was/am paid. At my old job I was paid an additional allowance of £83.33 gross each month whereas in my new job I am paid other allowances in total of £701.66 gross each month; in both cases the pension is not based on these allowances. I was paid apportions of these allowances in August based on the partial month I worked, likewise for the final pay in September I just received from my old job.

Also, to clarify, I realise there is nothing unusual about being paid thirteen times in a year when changing jobs. But in normal circumstances, the month of two pay days would equate to one paying for the first part of the month and the other paying for the second part and, I assume, not a big impact on tax contributions. In my case, however, due to the unusual 6th of the month payday cycle, it looks like I have been paid twice for two full months of work in the same month.