It is unfortunate that affordable formal jackets for women usually are lined using cheap polyester or acetate fabrics even when the jackets are made of quality materials like wool, linen, or silk. Synthetic fabrics like polyester and acetate are uncomfortable to wear. In my view, an acetate lining totally ruins a silk jacket.
Luckily it is no problem to replace a cheap synthetic fabric lining with a proper lining made of better quality fabrics which are more comfortable to wear. I usually use viscose satin or silk, and sometimes I combine them with cotton.
Here is a pink cotton jacket I found at a flea market. I am going to sew a new lining using more comfortable materials. First I carefully remove the old lining.
Then I take the half of the lining (left or right half) and cut all the seams open to separate the pattern pieces. I iron them and use them to cut new ones, adding seam allowances. I use the remaining half of the lining as an instruction on how to sew the pieces together, in case there are some complexities in the pattern (in this case, there are not, the lining is pretty basic).
Sometimes I take the easy way and make the new lining out of viscose satin or silk in a matching colour. But sometimes I am more playful. The jacket below has white viscose sleeves and sides while back and front are made of lightweight cotton fabric with lilac print. The purple piping was original and I did not remove it.
Here is another jacket with a new lining. This time I wanted to use whatever materials I had in stock, I just did not bother to go and buy anything. Again, I made the sleeves out of white viscose and the rest out of printed viscose fabric. I also used the grey pieces from the original lining.
The new lining to the pink jacket is still in progress. I did not buy any new fabrics this time either but used materials I already have. I used pink viscose satin and cotton print fabric to make the back side. I added dark green piping.
The new sleeves are made of white viscose fabric. The front was tricky, because I had only small pieces of fabric left. I made a paper pattern of the original front lining and cut it horizontally in three pieces. I used white viscose satin for the upper piece, the floral print cotton for the middle piece, and the pink viscose satin for the lower piece. I sew green piping between the pieces.
The jacket is now almost done, only the hem is still unfinished.
Here it the jacket finished, the hem sewn in place by hand and ironed.