* KKR is proposing to buy Hitachi Kokusai from parent Hitachi
* Elliott has 8.59 pct stake, Hitachi Kokusai's No 2 shareholder (Adds details, updates closing share price)
TOKYO, Oct 12 (Reuters) - U.S. hedge fund Elliott Management Corp has raised its stake in Japan's Hitachi Kokusai Electric to 8.59 percent from 7.11 percent, a regulatory filing showed, one of several such hikes since it first disclosed a stake in the firm last month.
The filing on Thursday comes a day after KKR & Co LP increased its offer price for the Hitachi Ltd unit to 2,900 yen ($25.80) a share from 2,503 yen, improving the bid after a third-party committee reporting to Hitachi Kokusai said it did not support the initial terms.
Elliott, known for buying stakes in firms in the middle of takeovers and seeking better deals for shareholders, has become Hitachi Kokusai's No.2 shareholder after Hitachi Ltd, which has a little over 50 percent.
Since Elliott's first disclosure of its stakeholding on Sept. 11, Hitachi Kokusai's share price has risen 5 percent and is now about 20 percent above KKR's original offer.
Elliott gradually amassed more shares each day over the past two weeks, the filing shows, bumping up its purchase on Wednesday when the Nikkei business daily reported KKR's plan for a higher offer.
Shares in Hitachi Kokusai fell 3 percent on Thursday to close at 3,020 yen, versus a 0.4 percent gain in the wider market.
Elliott has said the purpose of owning Hitachi Kokusai shares was "investment", but it has also noted it would "discuss matters such as important proposals depending on situations".
Hitachi Kokusai on Wednesday raised its full-year profit forecast by nearly 40 percent as the chip-making equipment maker is seeing strong capital investment by client semiconductor manufacturers.
$1 = 112.4100 yen Reporting by Chang-Ran Kim and Taiga Uranaka; Editing by Himani Sarkar and Stephen Coates