There is no LZ77 (that compression does not exists, it's name is LZ10, not LZ77), Huffman, RLE... header. The "header" is just 4 bytes. The first one indicates the type of compression (0x10, 0x11, 0x30, 0x24 or 0x28) and the other three bytes indicates the decompressed size. So, detecting compression just searching for a byte value (ie: searching 0x10 byte in a file) is not a very good idea... So many fake matching cases. The only way to know if a file is compressed is trying to decompress it. This method, with big files, with a lot of matching cases, is so time expensive (imagine a file of 2 MB of size filled with 0x10 byte... how many times Tinke will try to decode it?).
What Tinke can do is to unpack a file once you know its format (and if it's enough simply to use the automatic tool, else you will need to write a plug-in in .NET language). To do it, select the file and then click at "Open as -> Pack". Configure the parameters in that new window to match the pack format.
If you were right, and the file is unpacked (uncompressed) correctly, it will detect the subfiles as encoded (if they are) with LZ10 or whether compression have. Then you can decompress each of them, and if it's a supported image, open it.
Anyway, tell me exactly what game is (if possible GameCode too) and I will check the format and give you more info ;)