You can try increasing the voltage (multiple cells in series), doubling the wire back so you have 1/2 the length with two parallel strands, or getting a thicker wire such as AWG30. Doubling the wire back (two strands of half the length will give you double the power in each strand x 2 strands so 4x the power.
In general the power delivered to each strand is \$P = \frac{V^2A}{\rho L}\$ where L is the length of the wire, V is the voltage, A is the cross-sectional area of the wire and \$\rho\$ is a constant that depends on the alloy (actually it's not quite constant- it generally increases with temperature for metals and alloys).
The final temperature the wire will reach is a (often quite nonlinear) function of the power, and depends on the surface area of the wire, air currents, if it's in contact with anything, if it's wound in a coil so it heats itself etc.
In any case, you can see that increasing the voltage has a large effect on the power - it's squared so going from 3.6 to 11.1V will increase the power by almost 10:1. Decreasing the length has a reciprocal effect. And fattening up the wire increases the power (but also increases the surface area).
Also make sure you're not losing too much power in the connecting copper wires.
Of course you need to take care not to draw excessive current from then battery (use a protected battery to avoid possible fires or worse from the battery) and be careful not to overload or short the battery anyway and the wire itself could cause burns or start a fire. With proper precautions a bit of judicious experimentation can help get a feel for how it behaves.
