There are over 42,000 Lincolnshire Regiment service and pension records (for this regiment - and its antecedents) in various War Office series held at the National Archives. Clicking on the link will take you to the results on Findmypast but you will need a subscription or Pay-Per-View credits to actually view the records. Some of these records can also be viewed on-line on Ancestry although Findmypast has by far the most comprehensive service record collection.
Prior to 1st July 1881, the Lincolnshire Regiment was the 10th Regiment of Foot. It started a new regimental number series from this date, some samples from which I list below. Use this list as a guide to determine when your own ancestor might have joined this regiment - but note that this list is for regular enlistments only. Special Reserve and Territorial Force battalions operated their own distinct regimental number series.
Here are some sample army service numbers and corresponding joining dates for the (Regular) 1st and 2nd Battalions of the Lincolnshire Regiment.
124 joined on 13th October 1881
329 joined on 28th August 1882
584 joined on 2nd September 1883
918 joined on 5th November 1884
1364 joined on 18th October 1885
1620 joined on 15th October 1886
1959 joined on 15th June 1887
2168 joined on 9th May 1888
2362 joined on 14th June 1889
2480 joined on 8th January 1890
2700 joined on 12th February 1891
3129 joined on 16th June 1892
3490 joined on 23rd January 1893
3860 joined on 29th January 1894
4133 joined on 23rd March 1895
4506 joined on 30th January 1896
4720 joined on 1st May 1897
4916 joined on 4th January 1898
5305 joined on 24th March 1899
5584 joined on 26th January 1900
5839 joined on 15th February 1901
6078 joined on 27th January 1902
6310 joined on 19th January 1903
7030 joined on 16th May 1904
7561 joined on 31st May 1905
7700 joined on 20th February 1906
7886 joined on 7th January 1907
8411 joined on 27th January 1908
8615 joined on 13th January 1909
8902 joined on 9th April 1910
9198 joined on 9th August 1911
9455 joined on 12th October 1912
9607 joined on 23rd June 1913
9745 joined on July 6th 1914
As far as I can work out, the Lincolnshire Regiment had reached around 9762 by the time Britain declared war on Germany on 4th August 1914. The service battalions which began forming shortly afterwards, then followed the same numbering sequence that the regular battalions had maintained since 1881, and over the next four and a half years there was no distinction in numbering between men who enlisted for wartime service only and those who enlisted as career soldiers for Seven and Five.
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Also see my other posts regarding the Lincolnshire Regiment:
The 3rd (Special Reserve) Battalion, The Lincolnshire Regiment
The 4th Battalion, The Lincolnshire Regiment
The 5th Battalion, The Lincolnshire Regiment
The Lincolnshire Regiment - Service Battalions
The Lincolnshire Regiment - 10th Battalion - Grimsby Chums
Donald Banks - A Lincolnshire Terrier
The Lincolnshire Yeomanry
Search British Army WW1 Records HERE!