Hello Furkan,
Because of the close proximity of the card and phone, the phone is changing the resonance frequency of the card.
As example in free air the card is tuned to 14.2MHz. When beeing put in the case of the resonance increases to something like 19MHz. This effect is due to metal parts inside the phone.
This is affecting the power transfer between reader and card. The card will receive less power and can therefore not be read successfully.
Also the readers antenna geometry can play a role here.
Since you said, that the NFC06A1 can read the card in this condition, it should be also possible to read it with your device.
AAT (antenna tuning) can be a great help here, but can also make it a bit more complicated.
I would recommend to measure the smith chart of the reader in free air and 1cm distance to the card in the case.
If you are using AAT (like on the default NFC06A1) you have to power the board and disable the driver (Register 27h set to 0xXF => bit d_res =0xF).
If no AAT is used, the board can simply left unpowered.
The above mentioned application notes can be a great help to increase the output power of your device.
Please feel free to share your custom antenna design (including antenna parameters) as well as the above mentioned smith charts.
br Travis