During summer in the early 1960s, BR ran two return trains a day between Blackpool and Southport. These were regular services (Mon-Fri) appearing in the LMR Passenger Timetable, not specific one-off excursions.
Local coach operators in traditional seaside resorts have always earned a crust running mid-week day trips *from* the resort to nearby attractions - for example, Blackpool to The Lakes or Chester Zoo (and more recently niche destinations like Bury Market).
Probably in the 1960s, British Railways tried to get in on the act, using some of the spare rolling stock lying around mid-week pending the next Summer Saturday rush. Certainly the trains were timed for the mid-day leisure market, and Southport "boarders" could have a day out in Blackpool just as easily as someone staying in Blackpool could sample the delights of Southport.
From June to September 1960, there was:
- 10:38 SX Blackpool Central - Southport Chapel St.
- 18:00 SX Blackpool Central - Southport Chapel St.
- 10:00 SX Southport Chapel St - Blackpool Central.
- 18:30 SX Southport Chapel St - Blackpool Central.
Between Preston & Southport, I suspect these ran via the Burscough North Curve, rather than the West Lancs Line. All trains went via the Fylde coast, calling at Lytham, St Annes, etc.
Other options for a day out by train in summer 1960 were:
- 08:45 SX Blackpool Central - Lakeside
(as in today's Lakeside & Haverthwaite Railway, when it was still BR and intact to Ulverston)
- 09:50 SX Morecambe Promenade - Lakeside.
- 17:05 SX Lakeside - Morecambe Promenade.
- 18:20 SX Lakeside - Blackpool Central.
I mention these because they called at the Furness seaside resorts of Grange-over-Sands and Ulverston in both directions.
None of the above trains are listed as Second Class Only, so presumably you could splash out on First Class accommodation if you were so inclined.
More in line with the coast-to-coast gist of the OP's question:
- 09:40 SO Saltburn & 10:07 SO West Hartlepool to Blackpool Central & Southport Chapel St.
- 13:05 SO Blackpool Central & 13:05 SO Southport Chapel St to Saltburn & West Hartlepool
Running July & August 1960, via Blackburn, Burnley, Skipton, Ilkley, Harrogate & Northallerton.
Through carriages between Southport & Saltburn, Blackpool & Saltburn and Blackpool & West Hartlepool.