We extend the analysis of the MAXIMA-1 cosmic microwave background ( CMB ) data to smaller angular scales . MAXIMA , a bolometric balloon experiment , mapped a 124 deg ^ { 2 } region of the sky with 10′ resolution at frequencies of 150 , 240 and 410 GHz during its first flight . The original analysis , which covered the multipole range 36 \leq \ell \leq 785 , is extended to \ell = 1235 using data from three 150 GHz photometers in the fully cross-linked central 60 deg ^ { 2 } of the map . The main improvement over the original analysis is the use of 3′ square pixels in the calculation of the map . The new analysis is consistent with the original for \ell < 785 . For \ell > 785 , where inflationary models predict a third acoustic peak , the new analysis shows power with an amplitude of 56 \pm 7 \mu { \mbox { K } } at \ell \simeq 850 in excess to the average power of 42 \pm 3 \mu { \mbox { K } } in the range 441 < \ell < 785 .