Hot thin oil filters better than cool thicker oil.
I think every oil filter has a bypass valve that allows oil to bypass the filter element, either in the filter assembly or very near it, when the differential pressure exceeds the spec. Say the input pressure is 60 psi and the output is 58 psi...just two psid--pounds per square inch differential--so the bypass valve does not open and all the oil is being filtered. Let's say you have a WIX 51359 oil filter which as an internal bypass valve rated to begin opening at 8 psid and is fully open at 11 psid. Lets say that the cold oil enters the filter at 60 psi and is so thick that there is a pressure drop down to 51 psi at the outlet. With this 9 psi differential across the filter element we know that part of the oil is bypassing the filter. This is essential. Otherwise the filter would be torn apart. A filter that would withstand the high pressure drop of cold oil would have to have pores in the filter media so big that it could not filter finely enough to do any good. (We do not know the spec of the Suzuki OEM filter. Aftermarket filter makers reverse engineer OEM filters, but sometimes list a filter that is already in their inventory that is close enough to OEM to work well.) (All the oil pressure numbers above are plucked out of thin air for illustration purposes only.)
My Toyota Tundra has a coolant-cooled oil cooler in the filter base, and my Volvo turbo has an oil cooler in the radiator hot tank (ATF cooler in the cold tank). I think every diesel engine I have worked on has a coolant-cooled oil cooler. They give more steady oil temperatures than an air cooled cooler.