Baby’s body very warm but no fever

At 37.1°C it's technically not a fever yet, so Kool Fever patches and sponging are not really necessary at that point, though I understand the worry when the neck and back still feel warm to the touch. Since your baby is below 6 months and just recovering, I'd say it's worth calling or visiting the PD just to get a proper check, because very young babies can change quite quickly and the doctor can also advise you on what temperature range to watch out for going forward.