How we change what others think, feel, believe and do
Say you are not happy and then wait for the other person to improve their offer. Do not ask for something specific -- let them decide.
Shake your head and look sad, annoyed or distressed. Frown. Purse your lips. Send body language signals to reinforce your unhappy state.
Do not be woolly, weakening your statement with softening qualifiers like 'I can't say I'm happy' or 'I'm not that happy'.
When they do make an improved offer, say you are still not happy.
When they have run out of offers, you can ask for one more thing and then say you'll be happy with this.
You know, your company let me down and I'm really not happy at all.
I'm not happy with that...I'm still not happy...If you can add the whole package then I'll be happy.
There is a social rule that says you should not be unkind to others, and an indicator of this is that they are unhappy. Declaring your unhappiness invokes this rule and encourages the other person to improve their offer.
Being unhappy also hints that you may back out or escalate. This provides a second line of pressure on the other person to work hard to get you happy again.
Not saying what you want has two effects. If you ask for something, it looks like you are playing a game to get more. When you say you are not happy, you are not asking. Also, you may well be surprised when they give you more than you expected or wanted.
This is a useful method to use when you have received poor customer service, where representative often have a clear directive that customers should be happy and also have lot of scope to ensure this.