Xterm 256 colors

I mainly use one of my raspberry pi headlessly.
So, I’m using putty from one laptop to connect to it with ssh.
Everything works fine but I want to add some colours… and here is the issue:
my $TERM variable is only xterm and when I look at the documentation they say that we can check this using a command such as infocmp:

infocmp | grep colors
colors#256 , cols#80, it#8, lines#24, pairs#256,

I only get

colors #8, cols#80, it#8, lines#24, pairs#64,

Many sites discourage changing the TERM variable but I don’t find anything how to force it to 256 colours in a correct way.

I’ve updated putty to the last version and I tried many possible configurations but it’s still the same…

Do you have such a behavior? What can I do/check ?