Change the cartridge or phono stage

I think that you will get more warmth from a cart or phono stage, rather than from a cable swap. But do not assume that a TT is just like an amp or CD player. The signal level is much much lower. Which makes cables from your cart to phono stage the most important in your system.

That said, look for a warm cart or phono stage. Grado is one place to look on the cart side, available at a number of price points. The EAR phono stages are often described as warm, but that may be outside your budget. Maybe others know less expensive phono stages that meet your criteria.

I am not sure why everyone on this thread is obsessed with your set-up. I will assume that you have a cart that is properly set up and that any new cart will be set up properly by you or a dealer. Yes, this is very important.

Like many audio components, some carts and phono stages are warmer than others. Not knowing your phono stage or cart, I can't comment on the relative benefit of replacing one or the other. Though it is usually said that the transducers, which convert physical movement to electricity (carts) or electricity to physical movement (speakers) are the most important parts of the audio chain, and these may also exhibit the greatest variability.

You may get some good warm cart and phono stage recs in this thread, but there should also be a number of other threads with recs that offer suggestions.