By definition, the strongest region of a magnetic field developed by a solenoid exists in the middle of it. The field is perpendicular to the direction of the current windings, so in straight loops, the induced field would also be straight through the loops.

In most cases, the loops are near circular, if I change the shape of the loop, visualise an elongated loop, almost like a long rectangular one:

._______A___________
1|__________________|2
B

The above loop is standing on a flat surface, so sides 1 and 2 are perpendicular to the surface. Now lets say I keep side 1 in place and hold side 2 and twist the loop(NOT COMPLETELY AROUND) so that 2 is parallel to the surface. the entire loop would then have a kind of spiral shape.

My question is, would the induced B field be pointing towards me closer to 1, and upwards at side 2?

 

You can use the right hand rule to give the magnetic field around a conductor. The right thumb points in the direction of the current, and the fingers show the direction of the field.