I'd probably suggest across to Manchester, then XC down to the Southampton area, then across from there. On mobile so can't currently suggest any specific routes.
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A quick glance on National Rail Enquries is suggesting two options:
Huddersfield - Manchester Piccadilly
Manchester Piccadilly - Reading
Reading - Guildford
Guildford - Havant
Huddersfield - Manchester Piccadilly
Manchester Piccadilly - Milton Keynes Central
Milton Keynes Central - Clapham Junction (Southern service)
Clapham Junction - Havant
Your request to avoid the London area altogether probably rules out using Clapham Junction, so the first itinerary may suit best